Annals of University of Craiova, Math. Comp. Sci. Ser.
Volume 33, 2006, Pages 154–173
ISSN: 1223-6934
Regular graphs in which every pair of points is missed by
some longest cycle
Boris Schauerte and Carol T. Zamfirescu
Abstract. In Petersen’s well-known cubic graph every vertex is missed by some longest cycle.
Thomassen produced a planar graph with this property. Grünbaum found a cubic graph, in
which any two vertices are missed by some longest cycle. In this paper we present a cubic
planar graph fulfilling this condition.
2000 Mathematics Subject Classification. 05C38.
Key words and phrases. Planar, cubic, 3-connected, graph.
1. Introduction
The large group of automorphisms and the hypohamiltonicity of Petersen’s graph
are notorious. It is understandable that many other hypohamiltonian graphs have
been discovered, and generalizations of hypohamiltonicity have been proposed. Such
a generalization was given by T. Zamfirescu, who asked whether there exist graphs G
such that for any k vertices of G there is a longest cycle of G avoiding those vertices.
He called this property Ck [7] and asked for j-connected graphs with the property
Ck and as small as possible [4].
Among planar 2-connected graphs the best known example for k = 2 was presented
in [5] and has 138 vertices. Among 3-connected graphs the smallest example verifying
C2 published so far has 75 vertices [6], while among planar 3-connected graphs the
first example appeared in [6] and had 14818 vertices, and a smaller one appears in [3]
and has 4277 vertices. Neither one is regular.
For k = 1, Petersen’s graph is cubic, but non-planar. In 1978 C. Thomassen found
an infinite family of cubic planar hypohamiltonian graphs [2].
It is our main goal here to provide planar graphs which enjoy property C2 and are
also regular.
2. Two-Connected Graphs
We start with the following result concerning cubic planar graphs with property C1.
Theorem 1. There exists a cubic 2-connected planar graph on 30 vertices such
that any vertex is missed by some longest cycle.
Proof. The graph of Fig. 1, being a straightforward modification of an example
due to Thomassen and published in [6], enjoys the required properties.
Received: July 15, 2005.
154
REGULAR GRAPHS
155
Fig. 1
Theorem 2. There exists a cubic 2-connected planar graph on 250 vertices such
that any pair of vertices is missed by some longest cycle.
Proof. We construct the graph G in the following way. First, consider the graph
G1 of Fig. 2:
Fig. 2
Each of its vertices will be replaced by a graph G2 , see Fig. 3, respecting the
location of the arrow-marked edges.
Fig. 3
We obtain a graph G3 . The intersection of any longest cycle of G3 with a G2 -copy
is a path with 20 vertices if the arrow-marked edge of that copy is used, or with 24
vertices if the other two end-edges of the G2 -copy are used.
Suppose we intercalate between ai and ai+1 (indices mod 5) and between a′i and
′
ai+1 (indices mod 5) isomorphic copies of a graph with m vertices, and between ai
and a′i isomorphic copies of a graph with n vertices, thus obtaining G.
156
B. SCHAUERTE AND C. T. ZAMFIRESCU
Then each cycle of the type a1 a2 a′2 a′3 a3 a4 a5 a′5 a′4 a′3 a′2 a′1 a1 in G1 induces in G a
cycle of length 5 · 20 + 4 · 24 + 5m + 4n. Each cycle of type a1 a2 a3 a4 a5 a′5 a′4 a′3 a′2 a′1 a1
in G1 induces in G a cycle of length 8 · 20 + 2 · 24 + 8m + 2n.
Both types of cycles of G must be longest cycles. So we must have 5 · 20 + 4 · 24 +
5m + 4n=8 · 20 + 2 · 24 + 8m + 2n, which yields 2n=12 + 3m.
To choose a small example, we consider the situation m = 0 and n = 6, so we
intercalate nothing between ai and ai+1 and between a′i and a′i+1 , while between ai
and a′i (1 ≤ i ≤ 5) we intercalate the graph of Fig. 4:
Fig. 4
The resulting 2-connected graph has 250 vertices, verifies C2, and is cubic.
3. Three-Connected Graphs
The first example of a 3-connected graph such that any pair of vertices is missed
by some longest cycle was given by Grünbaum [1] in 1974. It has 90 vertices and is
cubic.
Theorem 3. There exists a cubic 3-connected planar graph on 9120 vertices such
that any pair of vertices is missed by some longest cycle.
Proof. We take Thomassen’s graph T from [2] (94 vertices, Fig. 5), open it up at
some vertex, and introduce it at every vertex of T .
We have to prove that every pair of edges in T is avoided by some longest cycle of
T [6]. This turned out to be a tedious task - therefore we worked using a computer.
At the end of the paper we provide a table which associates to every pair of edges a
cycle omitting it. It uses the following notation for edges (defined as pairs of numbers
corresponding to the vertices of T , see Fig. 5):
0 : (1, 0), 1 : (2, 0), 2 : (0, 92), 3 : (2, 3), 4 : (3, 4), 5 : (4, 5), 6 : (5, 6), 7 : (1, 6), 8 : (2, 7), 9 : (7, 8), 10 : (8, 9),
11 : (3, 9), 12 : (9, 10), 13 : (10, 11), 14 : (4, 11), 15 : (11, 12), 16 : (12, 13), 17 : (5, 13), 18 : (13, 14), 19 : (14, 15),
20 : (6, 15), 21 : (8, 16), 22 : (16, 17), 23 : (10, 17), 24 : (12, 18), 25 : (18, 19), 26 : (14, 19), 27 : (16, 23), 28 : (23, 24),
29 : (20, 24), 30 : (17, 20), 31 : (20, 21), 32 : (21, 22), 33 : (18, 22), 34 : (22, 25), 35 : (25, 26), 36 : (19, 26),
37 : (24, 27), 38 : (27, 28), 39 : (21, 28), 40 : (28, 29), 41 : (25, 29), 42 : (27, 31), 43 : (30, 31), 44 : (23, 30),
45 : (29, 32), 46 : (32, 33), 47 : (26, 33), 48 : (31, 34), 49 : (34, 35), 50 : (35, 36), 51 : (32, 36), 52 : (34, 39),
53 : (39, 40), 54 : (37, 40), 55 : (35, 37), 56 : (37, 41), 57 : (41, 42), 58 : (36, 42), 59 : (42, 43), 60 : (33, 43),
61 : (30, 38), 62 : (38, 39), 63 : (38, 45), 64 : (44, 45), 65 : (7, 44), 66 : (40, 46), 67 : (46, 47), 68 : (41, 47),
69 : (43, 48), 70 : (48, 49), 71 : (15, 49), 72 : (45, 50), 73 : (50, 51), 74 : (51, 52), 75 : (46, 52), 76 : (47, 53),
77 : (53, 54), 78 : (54, 55), 79 : (48, 55), 80 : (52, 56), 81 : (53, 56), 82 : (51, 57), 83 : (57, 58), 84 : (56, 58),
85 : (58, 59), 86 : (54, 59), 87 : (55, 63), 88 : (50, 60), 89 : (60, 61), 90 : (57, 61), 91 : (59, 62), 92 : (62, 63),
93 : (61, 64), 94 : (64, 65), 95 : (65, 66), 96 : (62, 66), 97 : (64, 68), 98 : (67, 68), 99 : (60, 67), 100 : (68, 71),
101 : (71, 72), 102 : (65, 72), 103 : (72, 73), 104 : (69, 73), 105 : (66, 69), 106 : (69, 70), 107 : (63, 70), 108 : (67, 74),
109 : (74, 75), 110 : (71, 75), 111 : (73, 76), 112 : (76, 77), 113 : (70, 77), 114 : (74, 79), 115 : (79, 80), 116 : (80, 81),
117 : (75, 81), 118 : (81, 82), 119 : (82, 83), 120 : (76, 83), 121 : (83, 84), 122 : (84, 85), 123 : (77, 85), 124 : (85, 86),
125 : (49, 86), 126 : (44, 78), 127 : (78, 79), 128 : (78, 87), 129 : (87, 88), 130 : (80, 88), 131 : (88, 89), 132 : (82, 89),
133 : (89, 90), 134 : (84, 90), 135 : (90, 91), 136 : (86, 91), 137 : (87, 92), 138 : (91, 93), 139 : (92, 93), 140 : (1, 93).
REGULAR GRAPHS
Fig. 5
157
158
B. SCHAUERTE AND C. T. ZAMFIRESCU
Now we associate to the numbers in the table the corresponding cycles:
1: 92,87,88,80,81,82,89,90,84,83,76,77,70,63,55,48,49,86,91,93,1,6,15,14,19,18,22,25,26,33,43,42,41,47,53,54,
59,62,66,69,73,72,65,64,61,60,67,68,71,75,74,79,78,44,45,50,51,57,58,56,52,46,40,37,35,36,32,29,28,21,20,24,27,31,
34,39,38,30,23,16,17,10,11,12,13,5,4,3,9,8,7,2,0
2: 92,87,78,79,74,67,60,50,51,52,56,53,47,46,40,39,38,45,44,7,2,3,4,5,6,15,14,13,12,11,10,9,8,16,17,20,21,22,
18,19,26,25,29,28,27,24,23,30,31,34,35,37,41,42,36,32,33,43,48,49,86,85,84,83,76,77,70,69,73,72,65,66,62,63,55,54,
59,58,57,61,64,68,71,75,81,80,88,89,90,91,93,1,0
3: 0,1,93,91,90,84,83,76,73,69,70,77,85,86,49,48,43,33,32,36,42,41,37,35,34,39,40,46,47,53,56,52,51,57,58,59,
54,55,63,62,66,65,72,71,75,74,67,68,64,61,60,50,45,38,30,31,27,28,29,25,26,19,18,22,21,20,24,23,16,17,10,11,12,13,
14,15,6,5,4,3,9,8,7,44,78,79,80,81,82,89,88,87,92
4: 0,2,7,44,45,38,39,34,31,30,23,24,27,28,29,32,36,35,37,40,46,52,56,53,47,41,42,43,33,26,25,22,21,20,17,16,8,
9,10,11,4,5,13,12,18,19,14,15,6,1,93,91,90,84,85,86,49,48,55,54,59,58,57,51,50,60,61,64,65,72,73,69,66,62,63,70,77,
76,83,82,89,88,80,81,75,71,68,67,74,79,78,87,92
5: 0,2,3,9,8,16,23,30,38,39,40,37,41,47,46,52,51,50,45,44,78,87,88,89,90,91,86,85,84,83,82,81,80,79,74,75,71,
72,73,76,77,70,69,66,65,64,68,67,60,61,57,58,56,53,54,59,62,63,55,48,49,15,14,13,12,18,19,26,25,22,21,28,29,32,33,
43,42,36,35,34,31,27,24,20,17,10,11,4,5,6,1,93,92
6: 0,2,7,44,78,79,74,67,60,61,57,58,59,62,66,69,70,63,55,54,53,56,52,51,50,45,38,30,23,24,20,17,16,8,9,10,11,
4,5,6,15,14,13,12,18,19,26,33,32,29,25,22,21,28,27,31,34,39,40,46,47,41,37,35,36,42,43,48,49,86,91,90,89,82,83,84,
85,77,76,73,72,65,64,68,71,75,81,80,88,87,92,93,1
7: 92,0,2,7,8,16,17,20,24,23,30,38,39,40,46,52,56,58,59,62,63,70,69,66,65,64,68,67,60,61,57,51,50,45,44,78,87,
88,80,79,74,75,71,72,73,76,77,85,84,83,82,89,90,91,86,49,48,55,54,53,47,41,37,35,34,31,27,28,21,22,25,29,32,36,42,
43,33,26,19,18,12,11,10,9,3,4,5,13,14,15,6,1,93
8: 0,1,93,91,90,84,83,76,73,69,70,77,85,86,49,48,43,33,32,36,42,41,37,35,34,39,40,46,47,53,56,52,51,57,58,59,
54,55,63,62,66,65,72,71,75,74,67,68,64,61,60,50,45,38,30,31,27,28,29,25,26,19,14,15,6,5,13,12,18,22,21,20,24,23,16,
8,9,10,11,4,3,2,7,44,78,79,80,81,82,89,88,87,92
9: 0,2,3,9,10,11,12,13,5,6,1,93,91,90,84,85,86,49,15,14,19,18,22,21,20,17,16,8,7,44,45,38,39,34,31,30,23,24,
27,28,29,25,26,33,32,36,35,37,40,46,52,56,53,47,41,42,43,48,55,54,59,58,57,51,50,60,61,64,65,72,73,69,66,62,63,70,
77,76,83,82,89,88,80,81,75,71,68,67,74,79,78,87,92
10: 0,2,3,9,10,11,4,5,13,12,18,19,26,33,43,48,49,15,6,1,93,91,86,85,77,76,73,69,70,63,55,54,53,47,41,42,36,32,
29,25,22,21,28,27,31,30,23,24,20,17,16,8,7,44,45,38,39,34,35,37,40,46,52,56,58,59,62,66,65,72,71,68,64,61,57,51,50,
60,67,74,75,81,82,83,84,90,89,88,80,79,78,87,92
11: 0,1,93,92,87,88,89,82,83,76,77,85,84,90,91,86,49,48,55,54,59,58,57,51,50,60,61,64,65,66,62,63,70,69,73,72,
71,68,67,74,75,81,80,79,78,44,7,8,16,17,20,24,23,30,38,39,34,31,27,28,21,22,25,29,32,36,35,37,40,46,52,56,53,47,41,
42,43,33,26,19,18,12,13,14,15,6,5,4,11,10,9,3,2
12: 0,2,7,8,9,3,4,5,13,14,19,26,25,29,28,27,24,20,21,22,18,12,11,10,17,16,23,30,31,34,39,38,45,44,78,79,80,81,
82,83,76,77,70,69,73,72,65,66,62,63,55,54,59,58,57,61,64,68,71,75,74,67,60,50,51,52,56,53,47,46,40,37,41,42,36,32,
33,43,48,49,15,6,1,93,91,86,85,84,90,89,88,87,92
13: 0,2,3,4,5,13,14,19,18,12,11,10,17,20,24,27,31,30,23,16,8,7,44,45,38,39,34,35,37,40,46,52,56,58,59,62,66,65,
72,71,68,64,61,57,51,50,60,67,74,75,81,82,83,84,90,89,88,80,79,78,87,92,93,91,86,85,77,76,73,69,70,63,55,54,53,47,
41,42,36,32,29,28,21,22,25,26,33,43,48,49,15,6,1
14: 0,2,7,44,45,38,30,23,16,17,10,9,3,4,11,12,13,5,6,1,93,91,86,85,84,90,89,88,80,81,82,83,76,77,70,63,55,54,59,
62,66,69,73,72,65,64,61,57,58,56,53,47,41,37,35,36,42,43,48,49,15,14,19,18,22,25,26,33,32,29,28,21,20,24,27,31,34,
39,40,46,52,51,50,60,67,68,71,75,74,79,78,87,92
15: 0,1,93,91,90,89,82,83,84,85,86,49,15,6,5,4,11,12,13,14,19,18,22,21,20,17,10,9,3,2,7,8,16,23,24,27,28,29,25,
26,33,43,48,55,63,62,66,69,70,77,76,73,72,65,64,61,60,50,51,57,58,59,54,53,56,52,46,47,41,42,36,35,37,40,39,34,31,
30,38,45,44,78,79,74,67,68,71,75,81,80,88,87,92
16: 93,92,87,88,89,90,91,86,49,48,55,54,53,56,58,59,62,63,70,69,66,65,64,68,67,74,75,71,72,73,76,77,85,84,83,82,
81,80,79,78,44,45,50,60,61,57,51,52,46,47,41,42,43,33,32,36,35,37,40,39,38,30,31,27,28,29,25,26,19,14,15,6,5,13,12,
18,22,21,20,24,23,16,17,10,11,4,3,9,8,7,2,0,1
17: 0,2,3,9,10,17,20,21,22,25,26,19,18,12,11,4,5,13,14,15,6,1,93,91,90,89,88,80,81,82,83,84,85,86,49,48,55,54,
53,56,58,59,62,63,70,77,76,73,69,66,65,72,71,68,64,61,57,51,52,46,47,41,37,40,39,38,30,31,34,35,36,42,43,33,32,29,
28,27,24,23,16,8,7,44,45,50,60,67,74,79,78,87,92
18: 92,87,88,80,81,82,83,84,90,91,93,1,6,5,13,14,15,49,86,85,77,76,73,72,71,75,74,79,78,44,45,50,51,57,61,60,
67,68,64,65,66,69,70,63,62,59,58,56,52,46,40,37,41,47,53,54,55,48,43,42,36,35,34,39,38,30,31,27,28,21,22,25,29,32,
33,26,19,18,12,11,4,3,9,10,17,20,24,23,16,8,7,2,0
19: 92,93,91,86,49,15,14,19,26,33,32,36,35,34,31,27,24,23,30,38,39,40,37,41,42,43,48,55,63,70,69,73,76,77,85,
84,83,82,89,88,87,78,79,80,81,75,74,67,60,61,64,68,71,72,65,66,62,59,54,53,47,46,52,56,58,57,51,50,45,44,7,2,3,4,
11,10,9,8,16,17,20,21,28,29,25,22,18,12,13,5,6,1,0
20: 0,1,93,91,90,89,82,83,84,85,86,49,15,14,13,5,4,3,2,7,8,9,10,11,12,18,19,26,25,22,21,20,17,16,23,24,27,28,
29,32,33,43,48,55,63,62,66,69,70,77,76,73,72,65,64,61,60,50,51,57,58,59,54,53,56,52,46,47,41,42,36,35,37,40,39,
34,31,30,38,45,44,78,79,74,67,68,71,75,81,80,88,87,92
21: 0,2,7,8,16,17,10,9,3,4,11,12,13,14,19,18,22,21,20,24,23,30,38,39,40,46,52,56,53,47,41,37,35,34,31,27,28,
29,25,26,33,32,36,42,43,48,49,15,6,1,93,91,86,85,77,70,69,73,76,83,84,90,89,82,81,75,74,67,60,61,64,68,71,72,65,
66,62,63,55,54,59,58,57,51,50,45,44,78,79,80,88,87,92
22: 0,2,7,44,78,87,88,89,90,84,85,77,76,83,82,81,80,79,74,75,71,72,73,69,70,63,55,54,59,62,66,65,64,68,67,60,
61,57,58,56,53,47,46,52,51,50,45,38,39,40,37,41,42,36,35,34,31,30,23,16,8,9,3,4,11,10,17,20,24,27,28,21,22,18,
12,13,5,6,15,14,19,26,25,29,32,33,43,48,49,86,91,93,1
23: 0,2,3,9,8,7,44,45,38,30,31,27,28,29,25,22,21,20,24,23,16,17,10,11,4,5,6,1,93,91,86,49,15,14,13,12,18,19,26,
33,32,36,35,34,39,40,37,41,42,43,48,55,54,59,58,56,53,47,46,52,51,50,60,61,64,65,66,62,63,70,69,73,72,71,68,67,74,
75,81,82,83,76,77,85,84,90,89,88,80,79,78,87,92
24: 0,1,93,91,86,85,77,76,73,69,70,63,55,54,53,47,41,37,35,34,31,30,23,24,27,28,29,25,26,33,32,36,42,43,48,49,
15,6,5,13,14,19,18,22,21,20,17,16,8,9,10,11,4,3,2,7,44,45,38,39,40,46,52,56,58,59,62,66,65,72,71,68,64,61,57,51,50,
60,67,74,75,81,82,83,84,90,89,88,80,79,78,87,92
25: 0,1,6,15,14,13,5,4,3,9,8,16,17,10,11,12,18,19,26,25,22,21,20,24,23,30,31,27,28,29,32,33,43,42,36,35,34,39,
38,45,50,60,61,57,51,52,46,40,37,41,47,53,56,58,59,54,55,48,49,86,85,84,83,82,89,90,91,93,92,87,88,80,81,75,71,72,
73,76,77,70,63,62,66,65,64,68,67,74,79,78,44,7,2
26: 92,93,1,6,5,4,11,12,13,14,15,49,86,85,84,90,89,88,87,78,79,80,81,82,83,76,77,70,69,73,72,71,75,74,67,68,64,
65,66,62,63,55,48,43,33,32,29,25,26,19,18,22,21,28,27,31,30,38,39,34,35,36,42,41,37,40,46,47,53,54,59,58,56,52,51,
57,61,60,50,45,44,7,8,16,23,24,20,17,10,9,3,2,0
27: 0,2,3,9,10,17,16,8,7,44,45,50,51,52,56,58,57,61,60,67,68,64,65,66,69,73,72,71,75,74,79,78,87,92,93,91,86,85,
84,90,89,88,80,81,82,83,76,77,70,63,62,59,54,53,47,46,40,39,38,30,23,24,20,21,28,27,31,34,35,37,41,42,36,32,29,25,
22,18,12,11,4,5,13,14,19,26,33,43,48,49,15,6,1
REGULAR GRAPHS
159
28: 0,1,6,5,13,12,18,19,14,15,49,48,55,54,53,56,58,59,62,63,70,69,66,65,64,68,71,72,73,76,77,85,86,91,93,92,87,
78,79,80,88,89,90,84,83,82,81,75,74,67,60,61,57,51,52,46,47,41,42,43,33,26,25,22,21,28,29,32,36,35,37,40,39,34,31,
27,24,20,17,10,11,4,3,9,8,16,23,30,38,45,44,7,2
29: 92,93,91,90,89,88,87,78,44,45,50,51,52,56,58,57,61,60,67,68,64,65,66,69,70,77,76,73,72,71,75,74,79,80,81,82,
83,84,85,86,49,48,55,63,62,59,54,53,47,46,40,39,38,30,31,34,35,37,41,42,43,33,32,29,28,27,24,23,16,8,7,2,3,9,10,17,
20,21,22,25,26,19,18,12,11,4,5,13,14,15,6,1,0
30: 0,2,7,8,9,3,4,5,13,14,19,18,12,11,10,17,16,23,24,20,21,22,25,26,33,32,29,28,27,31,30,38,39,34,35,36,42,43,
48,55,63,62,59,54,53,47,41,37,40,46,52,56,58,57,51,50,45,44,78,79,80,81,82,89,88,87,92,93,91,90,84,83,76,73,72,71,
75,74,67,60,61,64,65,66,69,70,77,85,86,49,15,6,1
31: 92,87,78,44,45,50,51,57,61,60,67,68,64,65,66,62,59,58,56,52,46,40,37,41,47,53,54,55,63,70,69,73,72,71,75,
74,79,80,81,82,89,90,84,83,76,77,85,86,91,93,1,6,5,13,14,15,49,48,43,42,36,35,34,39,38,30,31,27,28,21,22,25,29,32,
33,26,19,18,12,11,4,3,9,10,17,20,24,23,16,8,7,2,0
32: 0,2,3,9,10,11,4,5,6,15,49,48,55,54,53,47,41,37,40,46,52,56,58,59,62,63,70,77,76,73,69,66,65,64,68,71,75,
74,67,60,61,57,51,50,45,38,39,34,35,36,42,43,33,32,29,25,26,19,14,13,12,18,22,21,28,27,31,30,23,24,20,17,16,8,7,44,
78,79,80,81,82,83,84,85,86,91,90,89,88,87,92,93,1
33: 0,92,87,78,79,74,75,71,68,67,60,50,51,52,46,40,37,35,36,32,33,26,25,29,28,27,24,23,30,31,34,39,38,45,44,
7,2,3,9,8,16,17,20,21,22,18,19,14,13,12,11,4,5,6,15,49,48,43,42,41,47,53,56,58,57,61,64,65,72,73,69,66,62,59,54,55,
63,70,77,76,83,82,81,80,88,89,90,84,85,86,91,93,1
34: 0,2,7,8,16,23,30,38,39,40,37,41,47,46,52,51,50,45,44,78,87,88,89,90,91,86,85,84,83,82,81,80,79,74,75,71,
72,73,76,77,70,69,66,65,64,68,67,60,61,57,58,56,53,54,59,62,63,55,48,49,15,14,13,12,18,19,26,25,22,21,28,29,32,33,
43,42,36,35,34,31,27,24,20,17,10,9,3,4,5,6,1,93,92
35: 1,0,92,93,91,90,89,82,81,75,74,79,80,88,87,78,44,45,50,51,57,58,59,54,55,63,62,66,65,64,61,60,67,68,71,
72,73,69,70,77,76,83,84,85,86,49,15,14,19,18,22,25,26,33,43,42,36,32,29,28,21,20,24,27,31,34,35,37,41,47,53,56,52,
46,40,39,38,30,23,16,17,10,9,8,7,2,3,4,11,12,13,5,6
36: 0,2,3,9,8,7,44,45,38,39,34,31,30,23,24,27,28,29,32,36,35,37,40,46,52,56,53,47,41,42,43,33,26,25,22,21,
20,17,10,11,4,5,13,12,18,19,14,15,6,1,93,91,90,84,85,86,49,48,55,54,59,58,57,51,50,60,61,64,65,72,73,69,66,62,63,
70,77,76,83,82,89,88,80,81,75,71,68,67,74,79,78,87,92
37: 0,1,6,15,49,86,91,90,89,88,80,79,74,75,81,82,83,84,85,77,76,73,72,71,68,67,60,50,51,57,61,64,65,66,69,
70,63,62,59,58,56,52,46,47,53,54,55,48,43,42,41,37,40,39,34,35,36,32,33,26,19,14,13,5,4,11,12,18,22,25,29,28,21,
20,17,10,9,3,2,7,8,16,23,24,27,31,30,38,45,44,78,87,92
38: 92,87,78,44,7,2,3,4,5,13,14,19,18,12,11,10,9,8,16,17,20,21,22,25,26,33,43,42,36,32,29,28,27,24,23,30,31,
34,35,37,41,47,46,40,39,38,45,50,51,52,56,53,54,59,58,57,61,60,67,74,79,80,88,89,82,81,75,71,68,64,65,72,73,76,83,
84,90,91,86,85,77,70,69,66,62,63,55,48,49,15,6,1,93
39: 0,92,93,91,86,49,15,6,5,4,3,9,8,16,17,10,11,12,13,14,19,18,22,21,20,24,23,30,38,39,34,31,27,28,29,25,26,
33,32,36,35,37,40,46,47,41,42,43,48,55,54,53,56,52,51,57,58,59,62,63,70,77,85,84,90,89,82,83,76,73,69,66,65,72,71,
75,81,80,88,87,78,79,74,67,68,64,61,60,50,45,44,7,2
40: 0,1,6,5,4,3,9,8,16,17,10,11,12,18,22,25,29,32,33,26,19,14,15,49,48,43,42,36,35,37,41,47,53,56,58,57,61,
64,65,72,73,69,66,62,59,54,55,63,70,77,76,83,82,81,80,88,89,90,84,85,86,91,93,92,87,78,79,74,75,71,68,67,60,50,51,
52,46,40,39,34,31,27,28,21,20,24,23,30,38,45,44,7,2
41: 0,1,6,15,14,19,18,22,25,26,33,43,42,41,47,53,54,59,62,66,65,72,73,69,70,63,55,48,49,86,91,93,92,87,88,
89,90,84,85,77,76,83,82,81,75,71,68,64,61,60,67,74,79,78,44,45,50,51,57,58,56,52,46,40,37,35,36,32,29,28,21,20,24,
27,31,34,39,38,30,23,16,17,10,11,12,13,5,4,3,9,8,7,2
42:0,2,3,4,11,12,13,5,6,1,93,91,90,84,83,76,73,72,65,64,68,71,75,81,82,89,88,80,79,74,67,60,61,57,58,59,
54,53,56,52,51,50,45,38,39,34,35,37,40,46,47,41,42,36,32,29,25,26,33,43,48,55,63,62,66,69,70,77,85,86,49,15,14,19,
18,22,21,28,27,31,30,23,24,20,17,10,9,8,7,44,78,87,92
43: 1,6,5,13,14,19,26,33,43,42,41,37,40,46,47,53,54,55,48,49,86,91,90,89,88,87,78,79,80,81,82,83,84,85,77,76,
73,72,71,75,74,67,68,64,65,66,69,70,63,62,59,58,56,52,51,57,61,60,50,45,44,7,8,9,10,17,16,23,24,20,21,28,27,31,30,
38,39,34,35,36,32,29,25,22,18,12,11,4,3,2,0,92,93
44: 0,2,3,4,5,13,12,11,10,9,8,7,44,45,38,30,23,16,17,20,24,27,31,34,39,40,46,47,53,56,52,51,50,60,67,74,75,81,
80,79,78,87,88,89,82,83,76,73,72,71,68,64,61,57,58,59,54,55,63,62,66,69,70,77,85,84,90,91,86,49,48,43,42,41,37,35,
36,32,33,26,25,29,28,21,22,18,19,14,15,6,1,93,92
45: 0,1,6,15,14,19,26,25,22,18,12,13,5,4,11,10,9,3,2,7,8,16,17,20,21,28,29,32,33,43,48,49,86,85,77,76,73,72,71,
75,81,80,79,74,67,68,64,65,66,69,70,63,62,59,54,53,56,58,57,61,60,50,51,52,46,47,41,42,36,35,37,40,39,34,31,27,24,
23,30,38,45,44,78,87,88,89,82,83,84,90,91,93,92
46: 0,92,87,88,80,81,75,71,68,67,74,79,78,44,45,38,30,31,34,39,40,37,35,36,42,41,47,46,52,56,53,54,59,58,57,51,
50,60,61,64,65,72,73,76,77,70,69,66,62,63,55,48,43,33,32,29,28,27,24,23,16,17,20,21,22,25,26,19,14,13,12,11,10,9,8,
7,2,3,4,5,6,15,49,86,85,84,83,82,89,90,91,93,1
47: 7,2,3,4,11,12,13,5,6,15,14,19,18,22,25,29,28,21,20,17,10,9,8,16,23,24,27,31,30,38,45,50,60,61,64,68,67,74,
75,71,72,65,66,62,63,55,54,59,58,57,51,52,56,53,47,46,40,39,34,35,37,41,42,36,32,33,43,48,49,86,85,77,70,69,73,76,
83,84,90,91,93,1,0,92,87,88,89,82,81,80,79,78,44
48: 15,6,5,4,11,10,9,3,2,7,8,16,17,20,24,27,28,21,22,18,12,13,14,19,26,25,29,32,33,43,48,55,63,62,66,69,70,77,
76,73,72,65,64,61,60,50,51,57,58,59,54,53,56,52,46,47,41,42,36,35,37,40,39,34,31,30,38,45,44,78,79,74,67,68,71,75,
81,80,88,87,92,0,1,93,91,90,89,82,83,84,85,86,49
49: 0,92,87,88,80,79,78,44,45,50,51,57,58,59,54,55,63,62,66,65,72,71,68,64,61,60,67,74,75,81,82,89,90,84,83,76,
73,69,70,77,85,86,91,93,1,6,5,13,14,15,49,48,43,42,36,32,33,26,25,29,28,27,31,34,35,37,41,47,53,56,52,46,40,39,38,
30,23,24,20,21,22,18,12,11,4,3,9,10,17,16,8,7,2
50: 0,92,87,78,79,74,67,68,71,75,81,80,88,89,82,83,76,77,70,63,62,66,69,73,72,65,64,61,60,50,51,57,58,59,54,55,
48,49,86,85,84,90,91,93,1,6,15,14,13,5,4,11,12,18,19,26,33,43,42,41,47,53,56,52,46,40,37,35,36,32,29,25,22,21,28,
27,24,23,30,31,34,39,38,45,44,7,8,16,17,10,9,3,2
51: 0,92,87,88,89,82,81,80,79,78,44,7,2,3,9,8,16,23,30,38,45,50,60,61,64,68,67,74,75,71,72,65,66,62,63,55,54,59,
58,57,51,52,56,53,47,46,40,39,34,31,27,28,29,25,22,21,20,17,10,11,4,5,6,15,14,13,12,18,19,26,33,32,36,35,37,41,42,
43,48,49,86,85,77,70,69,73,76,83,84,90,91,93,1
52: 0,1,93,91,90,89,82,83,84,85,86,49,15,6,5,13,14,19,26,33,43,48,55,63,62,66,69,70,77,76,73,72,65,64,61,60,50,
51,57,58,59,54,53,56,52,46,47,41,42,36,32,29,28,27,24,20,21,22,18,12,11,4,3,2,7,8,9,10,17,16,23,30,31,34,35,37,40,
39,38,45,44,78,79,74,67,68,71,75,81,80,88,87,92
53: 0,2,3,4,5,6,1,93,91,90,84,85,86,49,15,14,13,12,11,10,9,8,7,44,45,38,39,34,31,30,23,16,17,20,24,27,28,29,25,
22,18,19,26,33,32,36,35,37,40,46,52,56,53,47,41,42,43,48,55,54,59,58,57,51,50,60,61,64,65,72,73,69,66,62,63,70,77,
76,83,82,89,88,80,81,75,71,68,67,74,79,78,87,92
54: 0,2,7,8,9,3,4,11,10,17,16,23,30,38,39,40,46,52,56,53,47,41,37,35,34,31,27,24,20,21,28,29,25,26,33,32,36,42,
43,48,49,15,14,19,18,12,13,5,6,1,93,91,86,85,77,70,69,73,76,83,84,90,89,82,81,75,74,67,60,61,64,68,71,72,65,66,62,
63,55,54,59,58,57,51,50,45,44,78,79,80,88,87,92
55: 0,1,93,91,90,84,83,76,73,69,70,77,85,86,49,48,43,33,32,36,42,41,37,35,34,39,40,46,47,53,56,52,51,57,58,59,
54,55,63,62,66,65,72,71,75,74,67,68,64,61,60,50,45,38,30,31,27,28,21,22,25,26,19,18,12,13,14,15,6,5,4,11,10,17,20,
24,23,16,8,9,3,2,7,44,78,79,80,81,82,89,88,87,92
160
B. SCHAUERTE AND C. T. ZAMFIRESCU
56: 0,92,87,88,80,81,75,71,68,67,74,79,78,44,45,38,30,31,34,39,40,37,35,36,42,41,47,46,52,56,53,54,59,58,57,51,
50,60,61,64,65,72,73,76,77,70,69,66,62,63,55,48,43,33,32,29,28,21,20,24,23,16,17,10,9,8,7,2,3,4,11,12,18,22,25,26,
19,14,13,5,6,15,49,86,85,84,83,82,89,90,91,93,1
57: 0,2,3,4,11,12,13,5,6,1,93,91,86,85,77,76,73,69,70,63,55,54,53,47,41,42,36,32,29,25,26,33,43,48,49,15,14,19,
18,22,21,20,24,27,31,30,23,16,17,10,9,8,7,44,45,38,39,34,35,37,40,46,52,56,58,59,62,66,65,72,71,68,64,61,57,51,50,
60,67,74,75,81,82,83,84,90,89,88,80,79,78,87,92
58: 0,1,93,91,90,84,83,76,73,69,70,77,85,86,49,48,43,33,32,36,42,41,37,35,34,39,40,46,47,53,56,52,51,57,58,59,
54,55,63,62,66,65,72,71,75,74,67,68,64,61,60,50,45,38,30,23,24,27,28,29,25,26,19,18,22,21,20,17,16,8,9,10,11,12,13,
14,15,6,5,4,3,2,7,44,78,79,80,81,82,89,88,87,92
59: 0,92,87,78,79,74,67,68,71,75,81,80,88,89,82,83,76,77,70,63,62,66,69,73,72,65,64,61,60,50,51,57,58,59,54,55,
48,49,86,85,84,90,91,93,1,6,15,14,13,5,4,11,12,18,19,26,33,43,42,41,47,53,56,52,46,40,37,35,36,32,29,25,22,21,28,27,
31,34,39,38,45,44,7,8,16,23,24,20,17,10,9,3,2
60: 1,93,91,86,85,77,70,69,73,76,83,84,90,89,82,81,75,74,67,60,61,64,68,71,72,65,66,62,63,55,54,59,58,57,51,50,
45,44,78,79,80,88,87,92,0,2,7,8,9,3,4,11,10,17,16,23,30,38,39,40,46,52,56,53,47,41,37,35,34,31,27,24,20,21,28,29,32,
36,42,43,48,49,15,14,19,26,25,22,18,12,13,5,6
61: 92,93,1,6,5,13,12,18,22,25,26,19,14,15,49,48,55,63,62,59,54,53,56,58,57,51,52,46,47,41,37,40,39,34,35,36,42,
43,33,32,29,28,21,20,17,16,23,24,27,31,30,38,45,50,60,61,64,65,66,69,70,77,76,73,72,71,68,67,74,75,81,82,83,84,85,
86,91,90,89,88,80,79,78,44,7,8,9,10,11,4,3,2,0
62: 93,91,90,84,85,77,70,63,55,54,53,56,52,46,47,41,42,43,48,49,15,6,5,4,3,2,7,8,9,10,11,12,13,14,19,18,22,21,
20,17,16,23,24,27,28,29,25,26,33,32,36,35,37,40,39,34,31,30,38,45,44,78,79,80,81,75,74,67,68,71,72,65,64,61,60,50,
51,57,58,59,62,66,69,73,76,83,82,89,88,87,92,0,1
63: 0,2,7,8,16,23,30,38,45,44,78,79,80,81,82,83,76,77,70,69,73,72,65,66,62,63,55,54,59,58,57,61,64,68,71,75,74,
67,60,50,51,52,56,53,47,46,40,37,41,42,36,35,34,31,27,24,20,17,10,9,3,4,11,12,18,19,26,25,22,21,28,29,32,33,43,48,
49,15,14,13,5,6,1,93,91,86,85,84,90,89,88,87,92
64: 1,6,15,14,19,26,33,43,48,49,86,85,84,83,82,81,75,74,67,68,71,72,65,64,61,60,50,51,57,58,59,62,66,69,73,76,
77,70,63,55,54,53,56,52,46,47,41,37,40,39,34,35,36,32,29,25,22,18,12,13,5,4,11,10,17,16,23,24,20,21,28,27,31,30,38,
45,44,7,8,9,3,2,0,92,87,78,79,80,88,89,90,91,93
65: 0,2,7,44,45,38,39,40,46,52,56,53,47,41,42,43,33,26,19,18,12,11,10,17,20,24,27,28,21,22,25,29,32,36,35,34,
31,30,23,16,8,9,3,4,5,13,14,15,6,1,93,91,90,84,85,86,49,48,55,54,59,58,57,51,50,60,61,64,65,72,73,69,66,62,63,70,
77,76,83,82,89,88,80,81,75,71,68,67,74,79,78,87,92
66: 92,87,78,44,45,50,51,52,46,40,37,41,47,53,54,55,63,70,77,76,83,82,81,75,71,68,64,65,72,73,69,66,62,59,58,
57,61,60,67,74,79,80,88,89,90,84,85,86,91,93,1,6,5,13,14,15,49,48,43,42,36,35,34,39,38,30,31,27,28,21,22,25,29,32,
33,26,19,18,12,11,4,3,9,10,17,20,24,23,16,8,7,2,0
67: 92,93,91,90,89,88,80,79,74,75,81,82,83,76,73,69,66,62,63,70,77,85,86,49,48,55,54,59,58,56,53,47,41,37,35,
34,31,27,28,21,22,25,29,32,36,42,43,33,26,19,18,12,11,10,9,3,4,5,13,14,15,6,1,0,2,7,8,16,17,20,24,23,30,38,39,40,
46,52,51,57,61,64,65,72,71,68,67,60,50,45,44,78,87
68: 0,2,7,8,16,23,30,38,45,44,78,79,80,81,82,83,76,77,70,69,73,72,65,66,62,63,55,54,59,58,57,61,64,68,71,75,74,
67,60,50,51,52,56,53,47,41,42,36,35,37,40,39,34,31,27,24,20,17,10,9,3,4,11,12,18,19,26,25,22,21,28,29,32,33,43,48,
49,15,14,13,5,6,1,93,91,86,85,84,90,89,88,87,92
69: 0,1,93,91,90,89,82,83,84,85,86,49,15,6,5,4,11,12,13,14,19,18,22,21,20,17,10,9,3,2,7,8,16,23,24,27,28,29,25,
26,33,32,36,35,37,41,42,43,48,55,63,62,66,69,70,77,76,73,72,65,64,61,60,50,51,57,58,59,54,53,56,52,46,40,39,34,31,
30,38,45,44,78,79,74,67,68,71,75,81,80,88,87,92
70: 0,2,3,9,10,11,4,5,6,15,14,13,12,18,19,26,33,32,29,25,22,21,28,27,31,30,23,24,20,17,16,8,7,44,78,79,80,81,75,
74,67,68,71,72,73,69,70,63,62,66,65,64,61,60,50,45,38,39,34,35,36,42,41,37,40,46,47,53,56,52,51,57,58,59,54,55,48,
49,86,91,90,84,85,77,76,83,82,89,88,87,92,93,1
71: 1,6,5,13,12,11,4,3,2,7,8,9,10,17,16,23,30,31,27,24,20,21,28,29,32,33,26,25,22,18,19,14,15,49,86,85,84,83,76,
77,70,69,73,72,71,68,67,60,61,64,65,66,62,63,55,48,43,42,36,35,34,39,40,37,41,47,46,52,56,53,54,59,58,57,51,50,45,
44,78,87,88,80,79,74,75,81,82,89,90,91,93,92,0
72: 92,87,78,44,45,50,60,67,74,75,81,80,88,89,82,83,84,90,91,93,1,6,15,14,13,5,4,3,9,10,11,12,18,19,26,33,43,42,
36,32,29,25,22,21,28,27,31,34,35,37,41,47,53,56,58,59,54,55,48,49,86,85,77,76,73,69,70,63,62,66,65,72,71,68,64,61,
57,51,52,46,40,39,38,30,23,24,20,17,16,8,7,2,0
73: 32,29,25,22,21,28,27,31,30,23,24,20,17,16,8,7,44,78,79,80,81,75,74,67,68,71,72,73,69,70,63,62,66,65,64,61,
60,50,45,38,39,34,35,37,40,46,47,53,56,52,51,57,58,59,54,55,48,49,86,91,90,84,85,77,76,83,82,89,88,87,92,93,1,0,2,
3,9,10,11,4,5,6,15,14,13,12,18,19,26,33,43,42,36
74: 31,27,24,20,21,28,29,32,33,26,25,22,18,19,14,15,49,86,85,84,83,76,77,70,69,73,72,71,68,67,60,61,64,65,66,
62,63,55,48,43,42,36,35,37,41,47,46,52,56,53,54,59,58,57,51,50,45,44,78,87,88,80,79,74,75,81,82,89,90,91,93,92,0,1,
6,5,13,12,11,4,3,2,7,8,9,10,17,16,23,30,38,39,34
75: 0,2,7,8,16,17,20,24,23,30,38,45,50,51,52,56,53,54,55,63,70,69,66,62,59,58,57,61,60,67,74,79,78,87,88,80,
81,75,71,68,64,65,72,73,76,77,85,84,83,82,89,90,91,86,49,48,43,42,36,35,37,41,47,46,40,39,34,31,27,28,21,22,25,29,
32,33,26,19,18,12,11,10,9,3,4,5,13,14,15,6,1,93,92
76: 1,6,15,14,19,18,22,25,26,33,43,48,55,54,53,56,52,51,50,60,67,68,64,61,57,58,59,62,63,70,77,85,86,91,90,84,
83,76,73,69,66,65,72,71,75,74,79,80,81,82,89,88,87,78,44,45,38,39,34,35,37,40,46,47,41,42,36,32,29,28,21,20,24,27,
31,30,23,16,17,10,11,12,13,5,4,3,9,8,7,2,0,92,93
77: 92,87,78,79,74,67,60,50,45,44,7,8,9,10,11,12,18,19,26,25,22,21,28,29,32,33,43,42,36,35,34,31,27,24,20,17,16,
23,30,38,39,40,37,41,47,46,52,56,53,54,59,58,57,61,64,68,71,75,81,80,88,89,82,83,76,77,70,69,73,72,65,66,62,63,55,
48,49,86,85,84,90,91,93,1,6,15,14,13,5,4,3,2,0
78: 92,93,1,6,15,14,19,18,22,25,26,33,43,42,41,47,53,56,58,57,61,60,67,68,64,65,66,69,70,63,62,59,54,55,48,49,
86,91,90,84,85,77,76,73,72,71,75,74,79,80,81,82,89,88,87,78,44,45,50,51,52,46,40,37,35,36,32,29,28,21,20,24,27,31,
34,39,38,30,23,16,17,10,11,12,13,5,4,3,9,8,7,2,0
79: 61,64,68,71,72,65,66,62,63,70,69,73,76,77,85,86,49,15,14,13,12,18,19,26,25,22,21,20,24,23,30,31,27,28,29,
32,33,43,48,55,54,59,58,56,53,47,46,40,37,41,42,36,35,34,39,38,45,44,7,2,3,9,8,16,17,10,11,4,5,6,1,0,92,93,91,90,
84,83,82,89,88,87,78,79,80,81,75,74,67,60,50,51,57
80: 62,66,69,73,72,65,64,61,60,67,68,71,75,74,79,78,44,7,8,9,10,17,16,23,24,20,21,22,25,26,33,32,29,28,27,31,
30,38,45,50,51,57,58,56,52,46,47,41,37,40,39,34,35,36,42,43,48,49,15,6,5,13,14,19,18,12,11,4,3,2,0,1,93,92,87,88,
80,81,82,89,90,91,86,85,84,83,76,77,70,63,55,54,59
81: 93,91,86,85,77,70,63,55,48,49,15,14,13,12,11,10,17,16,23,24,20,21,28,27,31,30,38,39,34,35,36,32,29,25,22,
18,19,26,33,43,42,41,37,40,46,47,53,56,52,51,57,58,59,62,66,69,73,76,83,84,90,89,82,81,75,74,67,68,71,72,65,64,61,
60,50,45,44,78,79,80,88,87,92,0,2,7,8,9,3,4,5,6,1
82:92,87,78,79,80,88,89,90,84,85,77,70,69,66,65,64,68,71,72,73,76,83,82,81,75,74,67,60,61,57,51,50,45,44,
7,8,9,10,11,12,18,19,26,25,22,21,28,29,32,33,43,42,36,35,34,31,27,24,20,17,16,23,30,38,39,40,37,41,47,46,52,56,53,
54,59,62,63,55,48,49,86,91,93,1,6,15,14,13,5,4,3,2,0
83: 0,1,6,5,4,3,2,7,44,45,38,39,40,37,35,34,31,30,23,24,27,28,29,25,26,19,18,22,21,20,17,16,8,9,10,11,12,13,
14,15,49,48,43,33,32,36,42,41,47,46,52,51,50,60,61,57,58,56,53,54,55,63,62,66,65,64,68,67,74,79,78,87,88,80,81,75,
71,72,73,69,70,77,76,83,82,89,90,84,85,86,91,93,92
REGULAR GRAPHS
161
84: 92,0,2,7,8,16,23,24,20,17,10,9,3,4,11,12,18,19,26,33,32,29,25,22,21,28,27,31,30,38,39,34,35,36,42,43,48,
49,15,14,13,5,6,1,93,91,86,85,84,90,89,82,83,76,77,70,63,55,54,53,47,41,37,40,46,52,56,58,59,62,66,69,73,72,65,64,
61,57,51,50,45,44,78,79,74,67,68,71,75,81,80,88,87
85: 92,93,1,6,5,13,12,18,22,25,26,19,14,15,49,48,55,63,62,59,54,53,56,58,57,51,52,46,47,41,37,40,39,34,35,36,
42,43,33,32,29,28,21,20,17,16,23,24,27,31,30,38,45,50,60,67,68,64,65,66,69,70,77,76,73,72,71,75,74,79,80,81,82,83,
84,85,86,91,90,89,88,87,78,44,7,8,9,10,11,4,3,2,0
86: 92,93,1,6,5,4,11,12,13,14,15,49,86,91,90,89,82,83,84,85,77,76,73,72,71,75,81,80,88,87,78,79,74,67,68,64,
65,66,69,70,63,55,48,43,33,32,29,25,26,19,18,22,21,28,27,31,30,38,39,34,35,36,42,41,37,40,46,47,53,54,59,58,56,52,
51,57,61,60,50,45,44,7,8,16,23,24,20,17,10,9,3,2,0
87: 93,1,6,15,14,19,26,25,22,18,12,13,5,4,11,10,17,16,23,30,31,27,24,20,21,28,29,32,33,43,42,36,35,34,39,38,
45,44,7,8,9,3,2,0,92,87,78,79,80,88,89,82,81,75,74,67,60,50,51,52,46,40,37,41,47,53,56,58,57,61,64,68,71,72,65,66,
62,59,54,55,48,49,86,85,77,70,69,73,76,83,84,90,91
88: 92,0,2,3,9,10,17,20,24,23,16,8,7,44,45,50,60,61,57,51,52,56,58,59,54,53,47,46,40,37,41,42,36,35,34,39,38,
30,31,27,28,21,22,18,19,26,25,29,32,33,43,48,55,63,62,66,65,72,71,68,67,74,75,81,80,79,78,87,88,89,82,83,76,73,69,
70,77,85,84,90,91,86,49,15,14,13,12,11,4,5,6,1,93
89: 92,93,1,6,5,13,12,18,22,25,26,19,14,15,49,48,55,63,62,59,54,53,56,58,57,51,52,46,47,41,37,40,39,34,35,36,
42,43,33,32,29,28,21,20,17,16,23,24,27,31,30,38,45,50,60,61,64,65,72,73,69,70,77,76,83,84,85,86,91,90,89,82,81,75,
71,68,67,74,79,80,88,87,78,44,7,8,9,10,11,4,3,2,0
90: 86,91,90,89,82,81,80,88,87,78,79,74,75,71,68,64,65,72,73,76,83,84,85,77,70,69,66,62,63,55,48,43,33,32,29,
25,26,19,18,22,21,28,27,31,30,38,39,34,35,36,42,41,37,40,46,47,53,54,59,58,56,52,51,57,61,60,50,45,44,7,8,16,23,
24,20,17,10,9,3,2,0,92,93,1,6,5,4,11,12,13,14,15,49
91: 78,87,88,89,82,83,84,90,91,86,85,77,76,73,69,66,65,72,71,75,81,80,79,74,67,68,64,61,60,50,45,38,30,31,27,
24,23,16,17,20,21,28,29,32,33,43,42,36,35,34,39,40,37,41,47,46,52,51,57,58,56,53,54,59,62,63,55,48,49,15,14,19,
26,25,22,18,12,13,5,6,1,93,92,0,2,3,4,11,10,9,8,7,44
92: 92,0,2,7,8,16,23,24,20,17,10,9,3,4,11,12,18,19,26,33,32,29,25,22,21,28,27,31,30,38,39,34,35,36,42,43,48,
49,15,14,13,5,6,1,93,91,86,85,84,90,89,82,83,76,77,70,63,55,54,53,47,41,37,40,46,52,56,58,59,62,66,69,73,72,65,
64,68,67,60,61,57,51,50,45,44,78,79,74,75,81,80,88,87
93: 92,87,78,79,80,88,89,82,81,75,74,67,60,50,51,52,46,40,37,41,47,53,56,58,57,61,64,68,71,72,65,66,69,70,63,
62,59,54,55,48,49,86,85,77,76,83,84,90,91,93,1,6,15,14,19,26,25,22,18,12,13,5,4,11,10,17,16,23,30,31,27,24,20,
21,28,29,32,33,43,42,36,35,34,39,38,45,44,7,8,9,3,2,0
94: 92,87,88,80,79,78,44,45,50,51,57,61,60,67,68,64,65,66,62,59,58,56,52,46,40,37,41,47,53,54,55,63,70,69,73,
72,71,75,81,82,89,90,84,83,76,77,85,86,91,93,1,6,5,13,14,15,49,48,43,42,36,35,34,39,38,30,31,27,28,21,22,25,29,
32,33,26,19,18,12,11,4,3,9,10,17,20,24,23,16,8,7,2,0
95: 92,0,2,3,9,10,17,20,24,23,16,8,7,44,45,50,60,61,57,51,52,56,58,59,54,53,47,46,40,37,41,42,36,35,34,39,38,
30,31,27,28,21,22,18,19,26,25,29,32,33,43,48,55,63,62,66,65,64,68,67,74,75,71,72,73,69,70,77,85,84,83,82,81,80,
79,78,87,88,89,90,91,86,49,15,14,13,12,11,4,5,6,1,93
96: 92,0,2,3,9,8,7,44,45,38,39,34,35,36,42,43,33,32,29,28,21,20,24,27,31,30,23,16,17,10,11,4,5,13,12,18,22,25,
26,19,14,15,6,1,93,91,90,84,85,86,49,48,55,54,59,62,63,70,69,66,65,64,61,57,58,56,53,47,41,37,40,46,52,51,50,60,
67,68,71,72,73,76,83,82,89,88,80,81,75,74,79,78,87
97: 92,87,88,80,79,74,67,60,50,51,52,56,53,47,46,40,39,38,45,44,7,2,3,4,5,6,15,14,13,12,11,10,9,8,16,17,20,21,
22,18,19,26,25,29,28,27,24,23,30,31,34,35,37,41,42,36,32,33,43,48,49,86,85,84,83,76,77,70,69,73,72,65,66,62,63,
55,54,59,58,57,61,64,68,71,75,81,82,89,90,91,93,1,0
Acknowledgement. We are indebted to T. Zamfirescu for some valuable suggestions.
References
[1] B. Grünbaum, Vertices missed by longest paths or circuits. J. Comb. Theory A 17, 31-38 (1974)
[2] C. Thomassen, Planar cubic hypohamiltonian and hypotraceable graphs. J. Comb. Theory B
30, 36-44 (1981)
[3] C. T. Zamfirescu and T. I. Zamfirescu, A planar hypohamiltonian graph with 48 vertices. J.
Graph Theory, to appear
[4] T. Zamfirescu, A two-connected planar graph without concurrent longest paths. J. Comb. Theory B 13, 116-121 (1972)
[5] T. Zamfirescu, L’histoire et l’état présent des bornes connues pour Pkj , Ckj , Pkj et Ckj . Cahiers
du CERO 17, 427-439 (1975)
[6] T. Zamfirescu, On longest paths and circuits in graphs. Math. Scand. 38, 211-239 (1976)
[7] T. Zamfirescu, Intersecting longest Paths or Cycles: A short Survey. Ana. Univ. Craiova 28,
1-9 (2001)
(Boris Schauerte) Hoddenfeld 21, 44149 Dortmund, Germany
E-mail address: [email protected]
(Carol T. Zamfirescu) Südwall 31, 44137 Dortmund, Germany
E-mail address: [email protected]
162
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35
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37
38
39
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41
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44
45
46
B. SCHAUERTE AND C. T. ZAMFIRESCU
0
+
38
38
1
4
9
1
39
5
4
7
4
1
14
1
4
7
5
1
10
5
1
5
4
1
12
5
4
1
4
1
5
1
4
9
7
1
7
1
4
7
1
4
1
17
9
1
1
37
+
38
3
15
8
20
2
3
2
15
2
3
15
2
8
20
2
8
15
19
3
8
2
2
8
2
2
35
2
3
48
19
20
2
19
15
3
19
2
48
20
2
19
3
2
15
2
37
37
+
6
6
16
13
6
11
6
11
6
13
27
13
6
13
6
16
13
28
16
13
6
41
16
6
6
13
38
16
6
27
6
16
6
13
6
13
16
6
13
22
6
16
16
11
3
19
22
5
+
4
14
1
3
3
4
7
4
1
14
1
4
7
3
1
12
14
1
18
4
1
12
3
4
1
4
1
6
1
4
3
6
1
3
1
3
6
1
4
1
3
3
1
4
25
2
26
26
+
9
4
6
5
4
9
4
5
15
9
4
17
5
4
10
5
23
5
4
9
27
5
4
5
4
23
5
27
4
9
6
4
6
5
4
6
4
4
5
15
9
4
5
13
1
26
19
26
+
21
8
9
8
9
8
16
14
9
8
18
21
8
21
9
14
8
8
9
8
18
9
14
9
8
18
14
18
8
18
9
8
14
8
18
14
9
14
8
8
28
6
19
2
5
5
2
19
+
20
10
4
7
4
1
17
1
4
7
21
1
10
20
1
13
4
1
12
7
4
1
4
1
7
1
4
12
7
1
7
1
4
7
1
4
1
17
12
1
7
19
3
11
19
3
19
19
+
3
2
11
2
3
15
2
6
20
2
8
15
20
3
8
2
2
8
2
2
22
2
3
6
47
6
2
6
15
3
47
2
6
20
2
6
3
2
11
8
13
2
5
5
2
10
2
18
+
5
9
13
3
17
3
5
13
3
9
10
5
3
5
9
3
27
3
9
5
9
3
5
27
5
3
10
9
3
5
3
10
5
9
5
3
3
10
9
25
1
6
7
15
1
6
18
18
+
14
2
5
14
2
4
24
2
4
24
5
14
5
2
2
8
2
2
5
2
8
5
14
4
2
6
4
6
5
2
6
4
2
5
8
2
4
10
19
3
5
5
3
12
5
3
5
31
+
13
13
14
7
10
7
11
9
10
9
14
13
7
9
27
7
7
13
9
14
7
14
7
9
7
9
7
13
9
7
13
9
7
15
9
7
11
13
2
37
61
2
10
2
4
2
31
31
+
13
35
2
4
13
2
4
13
19
20
8
2
2
8
2
2
13
2
8
6
19
4
2
6
4
6
13
2
6
4
2
6
8
2
4
12
25
1
6
61
3
1
6
3
12
1
3
61
+
33
1
5
12
3
1
12
5
1
5
33
1
12
3
25
1
33
1
5
1
5
3
23
1
3
1
3
22
1
12
1
3
3
1
13
29
22
5
5
29
22
5
38
5
7
5
31
97
+
34
34
17
15
14
15
14
14
15
33
14
27
17
21
14
15
14
18
14
17
15
18
14
18
14
15
18
14
15
14
15
15
15
Table 1. Rows 0-46; Columns 0-17
14
13
2
26
19
2
10
2
4
2
39
19
2
97
97
+
34
7
2
1
12
9
1
13
2
1
12
2
2
1
2
1
7
1
7
2
7
1
3
1
2
7
1
2
1
3
2
1
15
13
10
5
5
62
10
5
38
5
21
5
10
12
5
10
+
24
5
4
10
5
16
5
4
24
8
5
4
5
4
8
5
19
4
8
6
4
6
5
4
6
4
4
5
8
8
4
16
19
1
6
7
62
1
6
11
16
1
11
23
1
7
19
62
+
40
40
12
18
12
13
7
24
12
7
7
12
17
12
7
27
7
12
7
13
7
13
12
7
13
12
7
17
12
7
17
25
1
5
5
2
1
2
4
2
1
5
2
1
5
2
5
1
+
40
15
5
3
5
2
2
32
2
2
5
2
3
5
40
5
2
6
15
3
5
2
6
5
2
5
3
2
11
REGULAR GRAPHS
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
0
5
10
1
7
5
5
1
7
5
1
5
1
10
9
4
14
1
38
1
5
1
12
1
9
1
4
17
1
12
5
4
1
10
5
1
5
7
4
1
7
4
1
10
1
4
1
5
1
2
3
15
2
15
2
33
2
19
15
24
19
2
19
2
3
19
3
15
15
24
2
29
15
2
2
3
15
2
15
2
15
2
33
19
2
24
2
19
24
37
19
2
3
2
19
29
2
16
13
6
13
6
16
11
6
22
11
6
11
13
6
13
6
11
6
16
16
11
22
11
30
6
11
16
11
6
6
11
27
6
16
13
6
13
6
22
6
11
6
13
11
11
6
6
3
3
3
1
3
6
7
1
3
12
1
6
1
3
6
4
3
1
3
1
16
1
3
1
18
1
4
3
1
3
6
3
1
3
14
1
6
7
3
1
6
4
1
12
1
3
1
6
4
5
10
4
10
5
5
4
6
5
4
5
4
10
6
4
6
5
6
5
5
4
23
4
9
4
4
17
4
6
5
4
5
6
5
10
5
10
4
5
6
4
5
10
4
4
6
5
5
8
8
9
8
14
16
9
8
18
9
14
9
8
9
9
8
16
8
16
16
9
8
16
9
8
9
8
9
8
16
8
19
8
14
18
14
16
8
14
16
9
18
14
8
8
14
16
6
12
10
1
7
17
7
1
7
12
1
7
1
10
21
4
20
1
25
1
17
1
12
1
20
1
4
17
1
12
17
4
1
10
17
1
12
7
4
1
7
4
1
10
1
4
1
7
7
2
3
6
2
6
2
11
2
22
11
6
11
2
6
2
3
11
3
15
15
11
2
11
15
2
2
3
11
2
6
2
15
2
16
24
2
16
2
22
6
11
6
2
3
2
6
6
8
3
3
9
3
5
5
9
3
5
9
5
9
3
9
9
3
5
3
5
5
9
3
5
9
3
9
3
9
3
5
3
5
3
5
10
5
10
3
5
10
9
5
10
3
3
10
5
9
2
8
4
2
5
2
4
2
5
4
5
4
2
6
2
6
5
6
5
5
4
2
4
19
2
2
8
4
2
5
2
5
2
5
19
2
24
2
5
6
4
5
2
4
2
6
5
10
17
10
9
7
14
7
9
7
17
9
7
9
10
9
9
14
7
11
7
15
7
26
7
9
7
9
17
7
26
15
9
15
10
14
7
14
7
9
14
7
7
7
10
9
9
10
7
11
2
8
4
2
6
2
4
2
19
4
6
4
2
6
2
6
19
6
20
19
4
2
4
19
2
2
8
4
2
6
2
19
2
61
13
2
13
2
19
6
4
6
2
4
2
6
6
163
12
3
3
1
3
5
5
1
3
5
1
5
1
3
23
12
3
1
3
1
5
1
3
1
23
1
12
3
1
3
5
3
1
3
5
1
5
13
3
1
13
16
1
12
1
3
1
5
13
17
18
14
21
14
17
18
14
17
15
14
29
15
14
33
14
17
42
15
15
14
26
17
15
17
14
17
15
26
15
14
15
14
14
18
14
17
15
14
17
17
18
14
15
15
14
18
Table 2. Rows 47-93; Columns 0-17
14
2
3
1
2
20
2
1
2
12
1
7
1
2
9
2
3
1
3
1
20
1
2
1
9
1
2
3
1
2
20
2
1
2
25
1
2
7
2
1
7
7
1
2
1
2
1
7
15
5
8
4
8
5
5
4
6
5
4
5
4
8
6
4
6
5
6
5
5
4
8
4
19
4
4
8
4
6
5
4
5
6
5
10
5
10
4
5
6
4
5
10
4
4
6
5
16
12
13
12
7
17
7
12
7
12
13
7
29
12
18
12
20
7
25
7
17
7
12
7
18
7
12
17
7
12
17
12
20
12
17
7
12
7
12
27
7
7
7
12
20
12
13
7
17
2
3
6
2
5
2
11
2
5
11
5
11
2
6
2
3
5
3
5
5
11
2
5
15
2
2
3
11
2
5
2
5
2
5
26
2
23
2
5
6
11
5
2
3
2
6
5
164
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
B. SCHAUERTE AND C. T. ZAMFIRESCU
0
10
1
5
1
10
4
5
1
5
10
5
10
1
5
1
4
10
1
21
7
10
1
7
1
4
1
5
4
1
1
1
5
4
5
1
4
5
1
5
4
5
1
4
5
5
1
39
1
2
15
29
15
2
3
3
2
29
3
15
2
33
2
29
2
19
2
3
19
3
2
24
3
2
2
15
33
3
2
19
24
2
29
3
2
3
15
2
3
2
19
2
19
37
2
19
2
13
6
16
11
6
11
16
6
11
13
6
11
25
30
22
6
11
11
30
6
11
6
13
16
6
30
6
11
13
25
6
13
13
22
6
13
11
6
13
11
6
13
25
22
6
22
13
3
3
1
7
1
6
3
3
1
7
3
6
3
1
3
1
4
21
1
3
6
3
1
7
1
4
1
6
4
1
1
1
12
4
7
1
4
3
1
12
3
6
1
3
7
6
1
25
4
10
4
5
4
6
4
5
4
5
10
5
10
4
5
5
4
10
4
51
6
10
4
10
5
4
51
5
4
10
4
6
5
4
5
6
4
5
6
5
4
5
10
4
5
5
4
27
5
8
9
16
9
19
8
8
9
16
8
16
8
9
8
14
9
19
9
8
16
8
9
21
8
9
8
16
9
8
9
16
14
9
22
8
9
8
18
14
8
16
14
8
19
16
8
19
6
10
1
7
1
10
4
7
1
7
10
7
10
1
12
1
4
10
1
21
7
10
1
7
1
4
1
7
4
1
1
1
10
4
7
1
4
12
1
10
4
7
1
4
7
7
1
13
7
2
6
16
11
2
3
3
2
11
3
6
2
32
2
22
2
11
2
3
6
3
2
24
3
2
2
6
11
3
2
6
24
2
22
3
2
3
6
2
3
2
22
2
22
6
2
39
8
3
9
5
9
10
3
3
9
5
3
5
3
9
3
5
9
10
9
3
10
3
9
10
3
9
3
5
9
3
5
11
5
9
5
3
9
3
80
5
3
5
10
3
5
5
3
13
9
2
4
5
4
2
4
5
2
5
8
5
2
4
2
5
2
19
2
8
6
8
2
24
5
2
2
5
4
8
2
6
5
2
5
6
2
5
6
2
4
2
14
2
5
5
2
19
10
10
9
7
9
10
9
7
9
7
10
7
10
9
15
7
9
10
9
21
7
10
9
7
7
7
21
7
9
10
9
7
10
9
7
11
9
11
7
10
9
7
10
9
7
7
9
13
11
2
4
43
4
2
4
8
2
35
8
6
2
4
2
35
2
13
2
8
6
8
2
13
8
2
2
6
4
8
2
6
13
2
35
6
2
8
6
2
4
2
13
2
19
6
2
13
12
3
1
5
1
12
3
3
1
5
3
5
3
1
3
1
25
13
1
3
13
3
1
13
1
25
1
5
12
1
1
1
5
13
5
1
13
3
1
5
3
5
1
3
5
5
1
13
13
17
14
17
14
17
15
18
14
18
17
15
21
14
15
14
15
17
14
21
18
21
14
21
14
15
21
15
14
18
14
37
14
14
29
15
14
26
15
14
18
15
14
15
26
26
14
27
Table 3. Rows 94-140; Columns 0-17
14
2
1
7
1
2
3
3
1
7
3
7
2
1
2
1
2
13
1
3
7
3
1
7
1
2
1
7
9
1
1
1
12
2
7
1
2
3
1
2
3
2
1
2
7
7
1
13
15
8
4
5
4
6
4
5
4
5
8
5
8
4
5
5
4
10
4
8
6
8
4
10
5
4
8
5
4
8
4
6
5
4
5
6
4
5
6
5
4
5
10
4
5
5
4
19
16
12
20
7
20
12
20
7
12
7
13
7
12
17
12
7
17
13
12
30
7
12
17
7
7
7
30
7
12
13
12
7
12
13
7
12
13
12
7
12
18
7
12
17
7
7
12
13
17
2
6
5
11
2
3
3
2
5
3
5
2
32
2
5
2
11
2
3
6
3
2
21
3
2
2
5
11
3
2
6
5
2
5
3
2
3
6
2
3
2
21
2
5
5
2
39
REGULAR GRAPHS
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
18
13
1
38
45
3
1
67
3
10
1
3
10
1
31
10
10
1
1
+
10
9
1
8
4
1
8
10
4
1
4
1
10
1
4
8
10
1
8
1
4
10
1
4
1
8
8
1
19
27
4
11
22
4
12
67
4
12
39
11
4
11
22
4
12
11
4
67
+
43
12
13
10
15
12
10
10
12
15
12
10
27
10
12
10
13
10
13
12
10
13
12
21
15
12
10
20
13
10
5
5
15
10
5
18
5
6
5
10
6
5
10
5
6
5
10
78
+
14
5
9
9
19
5
9
5
9
14
5
14
5
9
18
9
18
5
9
18
5
9
5
18
9
28
21
30
1
38
35
2
1
2
3
2
1
3
2
1
31
2
21
1
1
1
78
78
+
36
20
1
12
3
36
1
20
1
42
1
20
3
23
1
3
1
3
42
1
12
1
3
3
1
22
19
2
6
7
2
19
2
4
2
6
11
2
6
7
2
62
6
2
45
4
6
2
+
8
15
8
5
36
5
15
8
5
37
5
8
18
13
8
5
8
18
5
15
5
8
8
13
23
27
1
5
5
3
1
5
3
5
1
3
14
1
5
14
5
1
1
1
12
5
1
7
+
2
8
2
2
44
2
8
6
19
4
2
6
4
6
19
2
6
4
2
6
8
2
4
24
13
10
7
7
35
10
7
38
10
7
41
10
21
7
10
10
7
35
10
23
7
21
7
7
+
46
2
2
1
2
1
26
1
46
2
47
1
3
1
2
26
1
2
1
3
2
1
25
25
1
6
39
2
1
2
4
2
1
41
2
1
98
2
14
1
1
1
4
6
1
2
1
41
+
49
19
12
19
8
22
19
46
8
19
49
8
19
8
22
45
12
19
8
8
27
26
13
3
11
19
3
10
16
3
10
21
3
10
3
98
10
10
11
12
3
11
10
3
11
3
10
98
+
2
5
2
3
5
53
5
2
6
49
3
5
2
6
5
2
5
3
2
7
27
13
10
6
7
18
10
6
11
10
6
11
10
6
7
10
10
6
96
10
11
6
31
6
7
7
6
10
+
48
2
21
6
19
4
2
6
4
6
19
2
6
4
2
6
48
2
4
28
30
3
38
39
3
21
67
3
67
21
3
38
3
38
39
21
39
96
3
39
95
3
39
3
21
39
3
96
+
51
1
5
1
5
12
51
1
51
1
12
22
1
12
1
48
12
1
29
27
1
11
22
2
1
2
4
2
1
11
2
1
22
2
12
1
1
1
4
95
1
2
1
23
1
11
11
95
+
50
50
19
4
2
19
4
51
19
2
50
4
2
19
15
2
4
165
30
13
10
11
19
17
10
17
11
10
21
11
10
11
35
10
10
11
17
10
11
10
21
11
17
10
17
10
10
21
11
+
50
1
23
3
23
1
3
1
3
27
1
12
1
3
3
1
31
25
2
6
39
2
39
2
4
2
6
41
2
6
38
2
38
6
2
38
4
6
2
2
17
38
2
94
6
38
2
17
+
53
5
22
6
13
6
5
53
6
5
22
5
18
44
7
Table 4. Rows 0-46; Columns 18-35
32
27
1
5
5
18
1
5
18
5
1
5
14
1
5
14
5
1
1
1
14
5
1
7
1
7
1
94
7
67
1
35
94
+
54
54
19
1
27
1
53
27
1
53
1
37
19
1
33
29
2
5
5
2
12
2
3
2
15
3
2
3
5
2
5
44
2
3
12
5
2
2
3
21
2
3
91
3
2
21
2
5
+
54
6
4
6
5
4
6
4
4
5
17
23
4
34
25
1
32
39
4
1
17
4
14
1
32
4
1
66
4
14
1
1
1
4
17
1
4
1
64
1
32
91
39
1
17
4
1
91
+
52
9
3
44
2
22
52
2
21
3
2
15
35
13
2
11
19
2
10
2
3
2
21
3
2
3
22
2
10
11
2
3
11
10
2
2
3
10
2
3
10
3
2
10
2
22
2
25
+
47
6
19
23
6
52
50
6
18
19
7
166
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
B. SCHAUERTE AND C. T. ZAMFIRESCU
18
8
8
1
8
14
16
1
8
19
1
14
1
8
9
4
8
1
8
1
16
1
8
1
9
1
4
8
1
8
16
4
1
8
14
1
14
10
4
1
10
4
1
10
1
4
1
16
19
12
10
12
10
15
21
10
21
12
10
21
33
10
21
10
15
21
32
12
15
10
12
32
15
43
10
39
10
12
15
12
15
10
33
10
12
10
12
27
10
27
21
10
15
12
10
27
20
5
18
9
35
5
5
9
14
5
9
5
9
20
9
9
14
5
42
5
5
9
19
5
9
43
9
26
9
26
5
9
5
14
5
18
5
18
9
5
18
9
5
14
9
9
14
5
21
3
3
1
3
14
16
1
3
12
1
14
1
3
14
12
3
1
3
1
16
1
3
1
20
1
12
3
1
3
16
3
1
3
14
1
12
16
3
1
16
16
1
12
1
3
1
16
22
5
8
15
8
5
5
13
8
5
13
5
28
8
18
13
8
5
8
5
5
13
8
5
15
8
13
8
13
8
5
8
5
8
5
13
5
13
8
5
13
17
5
13
8
8
13
5
23
2
8
4
2
6
2
4
2
19
4
6
4
2
6
2
6
7
6
7
19
4
2
4
9
2
2
8
4
2
6
2
19
2
33
7
2
7
2
19
6
4
6
2
4
2
6
6
24
2
3
1
2
14
2
1
2
26
1
14
1
2
9
2
3
1
3
1
15
1
2
1
9
1
2
3
1
2
15
2
1
2
14
1
2
24
2
1
24
9
1
2
1
2
1
26
25
8
8
12
8
22
16
12
8
12
16
32
16
8
19
12
8
16
8
12
16
32
8
16
19
8
12
8
19
8
16
8
19
8
16
19
12
16
8
19
16
16
19
12
8
8
19
16
26
2
3
6
2
5
2
10
2
5
10
5
11
2
6
2
3
5
3
5
5
7
2
5
18
2
2
3
7
2
5
2
5
2
5
7
2
7
2
5
6
7
5
2
3
2
6
5
27
2
10
4
2
6
2
4
2
19
4
6
4
2
6
2
6
7
6
7
19
4
2
4
9
2
2
25
4
2
6
2
19
2
25
7
2
7
2
19
6
4
6
2
4
2
6
6
28
5
13
1
12
5
5
1
14
5
1
5
1
12
14
12
14
1
22
1
5
1
12
1
35
1
12
28
1
12
5
12
1
12
5
1
5
13
12
1
13
28
1
12
1
12
1
5
29
2
37
4
2
15
2
4
2
17
4
17
4
2
9
2
15
17
38
15
15
4
2
4
9
2
2
17
4
2
15
2
15
2
17
19
2
17
2
19
17
4
19
2
4
2
19
29
30
3
3
1
3
14
16
1
3
12
1
14
1
3
14
12
3
1
3
1
16
1
3
1
23
1
12
3
1
3
16
3
1
3
14
1
12
16
3
1
16
16
1
12
1
3
1
16
31
5
10
6
7
5
5
10
6
5
10
5
11
10
6
10
6
5
6
5
5
7
22
5
18
6
10
26
7
6
5
11
5
6
5
7
5
7
6
5
6
7
5
10
11
11
6
5
Table 5. Rows 47-93; Columns 18-35
32
45
37
1
27
14
19
1
14
19
1
14
1
27
14
53
14
1
47
1
19
1
19
1
19
1
14
43
1
27
45
14
1
14
14
1
14
37
35
1
37
27
1
14
1
35
1
27
33
5
10
4
7
5
5
4
6
5
4
5
4
10
6
4
6
5
6
5
5
4
23
4
18
4
4
17
4
6
5
4
5
6
5
7
5
7
4
5
6
4
5
10
4
4
6
5
34
2
3
9
2
15
2
9
2
12
9
21
9
2
9
2
3
16
3
12
15
9
2
16
9
2
2
3
9
2
15
2
15
2
16
24
2
16
2
22
16
9
21
2
3
2
22
16
35
47
10
6
7
6
7
10
6
18
10
6
11
10
6
10
6
7
6
7
19
7
19
7
18
6
10
43
7
6
6
11
19
6
23
7
6
7
6
19
6
7
6
10
11
11
6
6
REGULAR GRAPHS
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
18
8
1
16
1
10
4
8
1
16
8
16
8
1
8
1
4
10
1
8
10
8
1
10
1
4
1
10
4
1
1
1
10
4
22
1
4
8
1
10
4
16
1
4
19
16
1
19
19
10
15
27
15
10
15
27
12
27
10
15
10
27
12
27
15
10
12
21
10
10
15
10
12
15
21
10
12
10
12
37
10
10
37
12
10
12
15
10
30
15
10
15
39
32
10
13
20
19
9
5
9
19
9
5
9
5
19
5
19
9
5
5
9
19
9
42
18
19
9
23
5
9
31
5
9
18
5
19
5
9
5
18
9
5
18
5
9
5
14
9
5
5
9
19
21
3
1
16
1
12
3
3
1
16
3
16
3
1
3
1
20
23
1
3
16
3
1
23
1
20
1
16
12
1
1
1
12
14
35
1
14
3
1
12
3
16
1
3
35
16
1
30
22
8
15
5
15
13
8
5
15
5
8
5
8
17
5
5
15
13
22
8
13
8
15
13
5
15
8
5
22
8
5
22
5
13
5
8
13
5
15
5
8
5
13
8
5
5
8
13
23
2
4
7
4
2
4
7
2
7
8
6
2
4
2
7
2
10
2
8
6
8
2
7
7
2
2
6
4
8
2
6
10
2
7
6
2
8
6
2
4
2
10
2
7
6
2
19
24
2
1
39
1
2
3
3
1
26
3
15
2
1
2
1
2
21
1
3
24
3
1
21
1
2
1
15
9
1
1
1
14
2
35
1
2
3
1
2
3
2
1
2
26
26
1
35
25
8
40
16
37
12
8
8
12
16
8
16
8
27
8
22
45
19
12
8
16
8
27
37
8
19
8
16
12
8
12
16
12
19
22
8
27
8
46
12
8
16
12
8
19
16
8
19
26
2
6
5
11
2
3
3
2
5
3
5
2
17
2
5
2
10
2
3
6
3
2
7
3
2
2
5
11
3
2
6
5
2
5
3
2
3
6
2
3
2
10
2
5
5
2
25
27
2
4
7
4
2
4
7
2
7
10
6
2
4
2
7
2
10
2
21
6
10
2
7
7
2
2
6
4
10
2
6
10
2
7
6
2
11
6
2
4
2
10
2
7
6
2
19
28
12
1
5
1
12
48
5
1
5
13
5
12
1
5
1
41
13
1
44
13
12
1
13
1
44
1
5
12
1
1
1
5
13
5
1
13
5
1
5
44
5
1
35
5
5
1
13
29
2
4
17
4
2
4
29
2
29
17
15
2
4
2
29
2
17
2
38
19
19
2
24
17
2
2
15
4
24
2
19
24
2
29
15
2
19
15
2
4
2
19
2
19
37
2
19
167
30
3
1
16
1
12
3
3
1
16
3
16
3
1
3
1
25
21
1
3
16
3
1
21
1
25
1
16
12
1
1
1
12
14
35
1
14
3
1
12
3
16
1
3
35
16
1
25
31
10
6
5
11
6
11
5
6
5
10
5
10
32
5
5
6
10
11
42
6
10
6
7
5
6
31
5
11
10
5
6
5
10
5
6
10
5
6
5
11
5
10
18
5
5
10
13
Table 6. Rows 94-140; Columns 18-35
32
19
1
27
1
19
43
27
1
27
19
37
19
1
35
1
41
19
1
47
19
19
1
35
1
19
1
19
14
1
1
1
14
14
35
1
14
19
1
14
19
19
1
35
19
37
1
19
33
10
4
5
4
6
4
5
4
5
10
5
10
4
5
5
4
10
4
30
6
10
4
7
5
4
30
5
4
10
4
6
5
4
5
6
4
5
6
5
4
5
10
4
5
5
4
13
34
2
9
16
9
2
3
3
2
16
3
15
2
9
2
22
2
21
2
3
16
3
2
21
3
2
2
15
9
3
2
16
12
2
22
3
2
3
15
2
3
2
12
2
22
16
2
39
35
10
6
7
11
6
11
7
6
7
10
6
10
27
47
7
6
10
11
47
6
10
6
7
7
6
31
6
11
10
27
6
10
10
7
6
10
11
6
10
11
6
10
18
7
6
10
19
168
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
B. SCHAUERTE AND C. T. ZAMFIRESCU
36
25
15
5
5
15
28
5
18
5
6
5
28
6
5
28
5
6
5
18
44
5
30
6
5
7
6
16
6
30
93
28
6
5
5
25
25
+
21
1
4
42
1
4
1
15
9
1
37
13
3
32
19
3
10
17
3
10
21
3
10
3
67
10
10
19
17
3
39
10
3
19
3
10
17
3
10
3
93
10
17
67
3
17
3
93
+
56
3
6
25
56
6
3
3
7
38
25
18
5
5
18
22
5
11
5
7
5
23
11
5
23
5
7
5
18
11
5
30
7
5
7
18
11
7
30
11
11
25
5
5
25
11
5
32
+
57
57
1
56
1
37
19
1
39
40
1
6
75
2
1
2
4
2
1
12
2
1
38
2
12
1
1
1
4
6
1
2
1
38
1
12
6
38
1
92
2
1
2
1
2
6
32
32
+
57
4
2
21
3
2
4
40
25
3
5
5
3
22
5
3
5
7
3
27
3
5
26
5
7
5
3
22
5
3
7
3
7
18
3
7
3
22
92
25
5
3
25
3
5
3
5
92
+
55
22
6
18
55
7
41
25
3
11
26
3
23
15
3
16
15
3
4
3
45
4
61
11
4
3
4
15
3
4
3
23
4
3
11
3
4
11
4
90
3
4
3
15
3
11
4
3
+
4
1
17
55
1
42
13
2
6
19
2
10
2
19
2
6
12
2
6
38
2
10
6
2
10
12
6
2
2
12
10
2
10
6
21
2
10
2
90
2
17
2
6
10
28
2
30
90
+
58
15
2
4
43
30
1
11
35
4
1
15
4
14
1
11
4
1
35
4
14
1
1
1
4
15
1
4
1
23
1
11
11
30
1
11
4
1
15
1
11
15
62
11
1
30
4
30
+
59
19
1
44
27
17
5
5
17
27
5
18
5
7
5
17
16
5
17
5
7
5
18
27
5
30
7
5
7
17
16
7
30
27
17
17
5
5
17
89
5
17
5
32
5
16
17
30
+
3
15
45
40
1
6
44
4
1
6
4
14
1
44
4
1
38
4
14
1
1
1
4
6
1
4
1
38
1
44
6
38
1
44
4
1
15
1
89
6
62
44
1
92
4
6
1
89
+
15
46
13
2
44
19
2
10
2
3
2
21
3
2
3
76
2
10
19
2
3
12
10
2
2
3
10
2
3
10
3
2
10
2
76
2
17
2
44
3
44
2
3
3
2
88
17
44
+
47
25
18
5
5
18
22
5
18
5
6
5
27
6
5
26
5
6
5
18
22
5
31
6
5
7
6
16
6
38
22
28
6
5
5
25
22
5
32
5
6
5
16
6
88
5
6
88
48
13
1
6
19
62
1
6
19
10
1
19
10
1
22
10
10
1
1
1
14
6
1
6
1
10
1
10
6
62
1
10
6
1
87
1
10
6
10
22
1
22
62
6
1
86
1
10
49
25
2
11
26
2
12
2
3
2
15
3
2
3
35
2
12
11
2
3
4
15
2
2
3
21
2
3
11
3
2
11
2
35
2
4
2
15
3
11
2
3
3
2
4
86
4
2
Table 7. Rows 0-46; Columns 36-53
50
19
1
11
19
3
1
15
3
23
1
3
4
1
35
4
21
1
1
1
4
15
1
4
1
21
1
3
11
3
1
11
4
1
3
1
3
15
3
11
1
3
3
19
1
30
1
3
51
13
2
37
44
2
10
2
44
2
84
12
2
12
37
2
10
37
2
10
12
10
2
2
12
10
2
10
10
44
2
10
2
14
2
14
2
37
10
37
2
76
84
2
14
17
14
2
52
19
1
5
5
18
1
5
18
5
1
5
27
1
5
19
5
1
1
1
22
5
1
6
1
7
1
19
6
21
1
19
6
1
5
1
19
5
19
5
1
5
84
6
1
5
1
19
53
25
4
7
7
4
23
7
4
16
7
11
4
11
7
4
27
7
4
18
4
7
76
4
7
7
4
11
7
39
4
11
4
7
87
4
11
7
17
7
4
7
4
17
4
7
4
17
REGULAR GRAPHS
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
36
47
13
1
13
14
21
1
14
30
1
14
1
13
9
4
14
1
38
1
15
1
44
1
9
1
4
28
1
38
15
4
1
13
14
1
14
13
4
1
13
4
1
13
1
4
1
28
37
3
3
6
3
6
7
10
3
18
10
6
11
3
6
10
3
7
3
7
16
7
3
7
18
3
10
3
7
3
6
3
26
3
16
7
6
7
3
27
6
7
6
10
3
3
6
6
38
5
13
1
13
5
5
1
14
5
1
5
1
13
14
13
14
1
47
1
5
1
19
1
19
1
13
28
1
44
5
14
1
13
5
1
5
13
35
1
13
28
1
13
1
35
1
5
39
2
3
4
2
15
2
4
2
12
4
17
4
2
9
2
3
16
3
12
15
4
2
4
9
2
2
3
4
2
15
2
15
2
16
24
2
16
2
29
16
4
21
2
3
2
24
16
40
22
10
6
7
6
7
10
6
18
10
6
11
10
6
10
6
7
6
7
22
7
22
7
18
6
10
26
7
6
6
11
26
6
22
7
6
7
6
22
6
7
6
10
11
11
6
6
41
5
13
1
13
5
5
1
14
5
1
5
1
13
14
4
14
1
25
1
5
1
29
1
20
1
4
17
1
29
5
4
1
13
5
1
5
13
4
1
13
4
1
13
1
4
1
5
42
2
58
4
2
15
2
4
2
12
4
17
4
2
9
2
15
17
22
12
15
4
2
4
9
2
2
17
4
2
15
2
15
2
17
24
2
17
2
22
17
4
22
2
4
2
22
22
43
5
58
1
7
5
5
1
6
5
1
5
1
27
6
59
6
1
6
1
5
1
19
1
19
1
11
28
1
6
5
11
1
6
5
1
5
7
6
1
6
7
1
14
1
11
1
5
44
3
3
15
3
15
16
18
3
17
15
17
16
3
18
59
3
16
3
15
15
18
3
16
15
3
15
3
15
3
15
3
15
3
16
18
23
16
3
29
16
16
18
17
3
3
31
16
45
2
3
9
2
15
2
9
2
12
9
21
9
2
9
2
3
16
3
12
15
9
2
16
9
2
2
3
9
2
15
2
15
2
16
19
2
16
2
19
16
9
19
2
3
2
19
16
46
60
10
1
7
15
7
1
7
43
1
7
1
10
60
4
15
1
11
1
15
1
27
1
15
1
4
28
1
27
15
4
1
10
28
1
27
7
4
1
7
4
1
10
1
4
1
7
47
+
3
12
2
5
2
12
2
5
16
5
16
2
60
2
3
5
3
5
5
25
2
5
20
2
2
3
20
2
5
2
5
2
5
26
2
16
2
5
16
16
5
2
3
2
22
5
169
48
6
+
16
3
18
16
10
3
18
10
18
16
3
18
10
3
16
3
16
16
10
3
16
18
3
10
3
10
3
16
3
26
3
16
10
23
10
3
30
10
16
18
10
3
3
10
16
49
25
86
+
12
6
16
1
6
12
1
6
1
12
6
4
6
1
6
1
15
1
12
1
9
1
4
16
1
6
6
4
1
6
14
1
6
16
4
1
6
4
1
12
1
4
1
6
Table 8. Rows 47-93; Columns 36-53
50
85
1
3
+
29
2
10
2
12
10
7
29
2
21
2
3
7
3
7
52
7
2
7
35
2
2
3
7
2
38
2
27
2
67
7
2
7
2
27
7
7
7
2
3
2
10
7
51
85
10
2
85
+
5
18
6
5
15
5
29
15
6
22
6
5
6
5
5
14
22
5
15
6
14
17
15
6
5
14
5
6
5
18
5
17
6
5
6
17
5
14
15
15
6
5
52
5
1
21
1
84
+
63
2
5
16
5
16
2
19
2
63
5
22
5
5
7
2
5
19
2
2
16
7
2
5
2
5
2
5
7
2
7
2
5
7
7
5
2
19
2
19
5
53
7
87
4
4
17
7
+
74
12
1
18
1
10
9
4
63
1
11
1
74
1
12
1
9
1
4
25
1
12
39
4
1
10
25
1
12
10
4
1
10
4
1
10
1
4
1
18
170
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
B. SCHAUERTE AND C. T. ZAMFIRESCU
36
13
1
28
1
13
4
30
1
28
13
15
13
1
15
1
4
13
1
21
13
13
1
13
1
4
1
13
4
1
1
1
13
4
35
1
4
30
1
13
4
15
1
4
35
38
1
13
37
3
6
7
11
6
3
3
6
7
3
6
3
25
3
7
6
10
11
3
6
3
6
7
3
6
3
6
11
3
25
6
10
10
7
3
10
3
6
10
3
6
10
3
7
6
3
25
38
13
1
5
1
13
45
5
1
5
13
5
13
1
5
1
41
13
1
44
13
13
1
13
1
19
1
5
14
1
1
1
5
13
5
1
13
5
1
5
19
5
1
35
5
5
1
13
39
2
4
16
4
2
3
3
2
16
3
15
2
4
2
29
2
17
2
3
16
3
2
21
3
2
2
15
4
3
2
16
12
2
29
3
2
3
15
2
3
2
12
2
29
16
2
25
40
10
6
7
11
6
11
7
6
7
10
6
10
27
26
7
6
10
11
42
6
10
6
7
7
6
31
6
11
10
26
6
10
10
7
6
10
11
6
10
11
6
10
18
7
6
10
27
41
13
1
5
1
13
4
5
1
5
13
5
13
1
5
1
4
13
1
30
13
13
1
13
1
4
1
5
4
1
1
1
5
4
5
1
4
5
1
5
4
5
1
4
5
5
1
13
42
2
4
17
4
2
4
22
2
22
17
15
2
4
2
22
2
17
2
38
22
12
2
24
12
2
2
15
4
24
2
22
12
2
22
12
2
12
15
2
4
2
12
2
22
38
2
29
43
19
1
5
1
6
11
5
1
5
19
5
11
1
5
1
6
11
1
21
6
11
1
7
1
6
1
5
11
1
1
1
5
14
5
1
14
5
1
5
11
5
1
35
5
5
1
19
44
3
15
16
15
17
3
3
15
16
3
15
3
17
3
18
15
17
23
3
16
3
15
23
3
15
3
15
23
3
15
16
31
17
29
3
17
3
15
16
3
15
23
3
26
16
3
29
45
2
9
16
9
2
3
3
2
16
3
15
2
9
2
33
2
19
2
3
16
3
2
21
3
2
2
15
9
3
2
16
12
2
37
3
2
3
15
2
3
2
12
2
19
16
2
19
46
10
1
7
1
10
4
7
1
7
10
7
10
1
15
1
4
10
1
38
7
10
1
7
1
4
1
7
4
1
1
1
10
4
7
1
4
11
1
10
4
7
1
4
7
7
1
13
47
2
20
5
20
2
3
3
2
5
3
5
2
17
2
5
2
17
2
3
16
3
2
60
3
2
2
5
12
3
2
16
5
2
5
3
2
3
20
2
3
2
12
2
5
5
2
25
48
3
42
16
23
10
3
3
32
16
3
16
3
25
3
18
25
10
23
3
10
3
18
10
3
25
3
10
23
3
25
16
10
10
31
3
10
3
18
10
3
16
10
3
26
16
3
13
49
12
1
16
1
6
4
16
1
11
39
6
11
1
12
1
4
11
1
39
6
11
1
28
1
4
1
6
4
1
1
1
12
4
45
1
4
11
1
12
4
6
1
4
39
6
1
28
Table 9. Rows 94-140; Columns 36-53
50
2
38
7
35
2
3
3
2
7
3
7
2
27
2
7
2
10
2
3
7
3
2
7
3
2
2
7
12
3
2
7
10
2
7
3
2
3
7
2
3
2
10
2
7
7
2
13
51
17
6
5
14
6
15
5
6
5
17
5
22
14
5
5
6
17
14
30
6
26
6
61
5
6
30
5
14
18
5
6
5
14
5
6
14
5
6
5
18
5
14
15
5
5
14
25
52
2
38
5
35
2
16
5
2
5
17
5
2
17
2
5
2
17
2
21
7
16
2
7
5
2
2
5
22
21
2
7
5
2
5
16
2
5
7
2
19
2
19
2
5
5
2
19
53
10
1
18
1
10
4
18
1
11
10
18
10
1
12
1
4
10
1
30
10
10
1
10
1
4
1
10
4
1
1
1
10
4
31
1
4
11
1
10
4
25
1
4
26
11
1
13
REGULAR GRAPHS
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
54
13
10
6
7
17
10
6
18
10
6
16
10
6
7
10
10
6
17
10
39
6
31
6
7
7
6
10
6
39
31
10
6
7
83
17
10
6
10
7
6
7
16
6
37
7
6
10
55
19
1
5
5
29
1
5
19
5
1
5
14
1
5
14
5
1
1
1
14
5
1
19
1
41
1
19
19
30
1
19
41
1
5
1
19
5
19
5
1
5
45
19
1
5
1
19
56
35
2
6
35
2
12
2
3
2
6
3
2
3
35
2
12
6
2
3
4
6
2
2
3
21
2
3
6
3
2
11
2
35
2
4
2
6
3
11
2
3
3
2
4
39
4
2
57
13
10
7
7
17
10
7
18
10
7
16
10
16
7
10
10
7
17
10
23
7
31
7
7
7
17
10
7
67
23
10
17
7
26
17
10
7
10
7
32
7
16
10
23
7
67
10
58
27
2
5
5
2
12
2
38
2
6
5
2
6
5
2
5
6
2
38
12
5
2
2
5
23
2
12
6
38
2
23
2
5
2
14
2
5
76
5
2
5
23
2
14
5
6
2
59
13
1
7
7
18
1
7
18
10
1
19
10
1
7
10
10
1
1
1
26
7
1
7
1
7
1
10
7
30
1
10
41
1
26
1
10
7
10
7
1
7
26
10
1
7
1
10
60
25
14
5
5
17
14
5
45
5
25
5
14
16
5
14
5
16
5
28
14
5
78
25
5
23
14
16
16
67
14
17
17
5
5
14
22
5
17
5
14
5
16
17
14
5
14
17
61
13
2
6
22
2
10
2
3
2
6
3
2
3
22
2
10
6
2
3
12
6
2
2
3
10
2
3
6
3
2
10
2
14
2
14
2
6
3
22
2
3
3
2
14
27
6
2
62
19
1
5
5
15
1
5
19
5
1
5
27
1
5
19
5
1
1
1
26
5
1
15
1
38
1
19
19
30
1
19
15
1
5
1
19
5
19
5
1
5
15
19
1
5
1
19
63
25
2
11
22
2
12
2
3
2
21
3
2
3
22
2
12
11
2
3
4
25
2
2
3
21
2
3
11
3
2
11
2
14
2
4
2
25
3
11
2
3
3
2
4
78
4
2
64
28
15
5
5
15
28
5
38
5
6
5
17
6
5
17
5
6
5
28
39
5
38
6
5
28
6
16
6
38
71
17
6
5
5
17
71
5
17
5
6
5
15
6
15
5
6
17
65
27
2
6
26
2
12
2
3
2
6
3
2
3
29
2
12
6
2
3
12
6
2
2
3
38
2
3
6
3
2
44
2
27
2
27
2
6
3
26
2
3
3
2
70
27
6
2
171
66
13
1
7
7
4
1
7
4
10
1
11
4
1
7
4
10
1
1
1
4
7
1
4
1
7
1
10
7
21
1
10
4
1
15
1
10
7
10
7
1
7
4
10
1
7
1
10
67
25
3
16
26
3
26
16
3
16
25
3
17
3
67
17
28
16
17
3
26
16
3
25
3
28
17
3
16
3
26
17
17
67
3
17
3
16
3
16
77
3
3
17
61
16
67
3
Table 10. Rows 0-46; Columns 54-71
68
13
2
11
44
2
10
2
4
2
15
11
2
11
37
2
10
11
2
10
4
10
2
2
12
10
2
10
10
44
2
10
2
14
2
4
2
15
10
11
2
45
4
2
4
28
4
2
69
25
1
6
7
2
1
2
3
2
1
3
2
1
7
2
76
1
1
1
4
6
1
2
1
7
1
3
6
3
1
11
2
1
2
1
2
6
3
7
1
3
3
2
1
7
1
2
70
19
9
26
19
9
19
15
9
18
15
19
9
15
35
9
76
19
9
18
9
15
30
9
18
23
9
19
18
30
9
19
9
18
15
9
19
15
19
18
9
18
9
19
9
18
9
19
172
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
B. SCHAUERTE AND C. T. ZAMFIRESCU
54
6
6
83
39
10
6
7
+
65
65
6
29
2
6
2
3
7
3
7
74
7
2
7
35
2
2
3
7
2
6
2
27
2
14
7
2
7
2
14
6
7
6
2
3
2
6
6
55
5
1
83
1
14
1
27
83
+
65
5
19
12
18
12
23
5
22
5
5
18
12
5
18
17
12
17
18
12
5
12
5
12
5
18
5
17
12
5
17
17
5
12
19
12
19
5
56
6
6
2
3
2
6
4
6
82
+
73
1
10
9
4
15
1
11
1
15
1
73
1
9
1
4
16
1
39
15
4
1
10
16
1
33
10
4
1
10
4
1
10
1
4
1
16
57
7
10
23
23
10
7
7
7
82
82
+
64
64
6
24
6
5
6
5
5
7
73
5
18
6
14
17
7
6
5
14
5
6
5
7
5
7
6
5
6
7
5
14
21
21
6
5
58
5
6
2
23
2
5
23
6
5
2
23
+
64
9
4
23
1
11
1
16
1
19
1
9
1
4
16
1
29
16
4
1
33
16
1
23
16
4
1
16
4
1
33
1
4
1
16
59
7
1
26
1
10
1
7
7
1
66
7
27
+
70
2
3
26
3
12
15
10
2
70
15
2
2
3
10
2
15
2
15
2
22
10
2
10
2
22
10
27
22
2
3
2
10
22
60
5
14
23
23
14
5
16
16
5
66
16
5
66
+
9
6
18
6
18
19
9
19
70
9
6
9
39
9
6
6
9
19
6
14
18
6
18
6
14
6
9
6
14
9
9
6
6
61
6
6
2
3
2
6
27
6
14
2
10
2
10
14
+
71
71
22
12
22
4
2
4
9
2
2
25
4
2
38
2
38
2
22
10
2
10
2
22
10
4
22
2
4
2
10
22
62
5
1
15
1
77
1
27
81
1
15
26
5
1
5
27
+
71
3
15
15
14
3
28
15
3
14
3
15
3
6
3
15
3
14
37
6
23
3
14
6
23
6
14
3
3
6
6
63
22
14
2
3
2
21
4
81
14
2
23
2
80
14
2
81
+
11
1
5
1
19
1
18
1
11
16
1
26
5
11
1
21
5
1
5
7
11
1
7
7
1
17
1
11
1
5
64
5
6
15
15
17
5
16
6
5
6
16
5
80
5
6
5
80
+
75
22
11
3
11
42
3
11
3
11
3
6
3
38
3
22
32
6
25
3
22
6
11
6
85
3
3
6
6
65
6
6
2
3
2
6
27
6
27
2
26
2
26
79
2
26
2
6
+
5
1
12
1
15
1
12
16
1
12
5
12
1
12
5
1
5
7
12
1
7
7
1
12
1
12
1
5
66
7
1
4
1
10
1
4
7
1
4
7
77
1
79
10
1
4
15
79
+
68
19
5
15
16
15
16
15
68
5
22
5
22
5
19
5
16
15
5
16
16
5
17
15
15
19
5
67
16
81
3
3
17
28
16
16
61
3
16
77
26
16
3
26
3
16
3
77
+
69
1
9
1
4
25
1
68
69
4
1
10
14
1
14
7
4
1
7
4
1
10
1
4
1
7
68
28
10
2
4
2
28
4
10
14
2
10
2
10
14
2
15
2
15
2
4
28
+
29
19
2
2
3
19
2
69
2
19
2
22
19
2
23
2
19
37
23
19
2
3
2
19
22
Table 11. Rows 47-93; Columns 54-71
69
6
1
2
1
2
1
4
6
1
2
7
2
1
16
2
1
2
6
2
1
3
2
+
35
1
4
16
1
29
5
4
1
35
5
1
5
7
4
1
7
4
1
17
1
4
1
5
70
18
19
9
9
37
18
9
18
19
9
18
23
18
23
35
15
9
15
26
9
26
9
76
+
76
9
26
9
26
15
9
15
35
23
18
23
18
9
19
18
9
18
37
9
9
19
18
REGULAR GRAPHS
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
54
2
6
7
14
2
3
3
2
7
3
6
2
14
2
7
2
21
2
3
6
3
2
7
3
2
2
6
14
3
2
6
14
2
7
3
2
3
6
2
3
2
14
2
7
6
2
27
55
12
64
5
23
12
23
5
12
5
17
5
12
17
5
5
17
17
12
30
18
12
17
23
5
19
30
5
12
18
5
19
5
17
5
12
17
5
18
5
18
5
12
17
5
5
12
19
56
10
1
16
1
10
4
16
1
11
10
15
10
1
15
1
4
10
1
39
10
10
1
10
1
4
1
10
4
1
1
1
10
4
42
1
4
11
1
10
4
15
1
4
39
11
1
13
57
17
6
5
14
6
25
5
6
5
17
5
21
14
5
5
6
17
14
21
6
21
6
7
5
6
21
5
14
18
5
6
5
14
5
6
14
5
6
5
18
5
14
17
5
5
14
25
58
19
1
16
1
19
4
16
1
11
19
16
11
1
29
1
4
11
1
39
11
11
1
23
1
4
1
16
4
1
1
1
28
4
29
1
4
11
1
16
4
16
1
4
19
11
1
19
59
2
15
27
15
2
3
3
2
22
3
15
2
27
2
22
2
10
2
3
10
3
2
10
3
2
2
10
12
3
2
22
10
2
22
3
2
3
15
2
3
2
10
2
22
26
2
13
60
19
6
18
9
6
9
18
6
18
19
6
19
9
21
14
6
19
9
21
6
19
6
21
14
6
21
6
9
18
9
6
14
9
31
6
9
19
6
14
9
6
14
9
19
6
9
19
61
2
4
32
4
2
4
22
2
22
10
25
2
4
2
22
2
10
2
38
10
10
2
10
12
2
2
10
4
10
2
22
10
2
22
12
2
12
25
2
4
2
10
2
22
32
2
13
62
3
6
28
14
6
3
3
6
23
3
6
3
14
3
14
6
23
14
3
6
3
6
23
3
6
3
6
14
3
14
6
14
14
37
3
14
3
6
14
3
6
14
3
44
6
3
28
63
17
1
5
1
17
11
5
1
5
17
5
11
1
5
1
17
11
1
21
7
11
1
7
1
7
1
5
11
1
1
1
5
17
5
1
17
5
1
5
11
5
1
17
5
5
1
19
64
3
6
32
11
6
3
3
6
11
3
6
3
25
3
22
6
11
11
3
6
3
6
38
3
6
3
6
11
3
25
6
32
75
22
3
38
3
6
22
3
6
22
3
22
6
3
25
65
12
1
5
1
12
15
5
1
5
21
5
12
1
5
1
15
21
1
21
7
12
1
7
1
7
1
5
12
1
1
1
5
75
5
1
31
5
1
5
18
5
1
15
5
5
1
29
173
66
17
15
5
15
17
15
5
15
5
17
5
19
17
5
5
15
17
22
62
16
16
15
23
5
15
71
5
22
28
5
16
5
17
5
15
17
5
15
5
19
5
19
15
5
5
15
19
67
10
1
7
1
10
4
7
1
7
10
7
10
1
21
1
4
10
1
21
7
10
1
7
1
4
1
7
4
1
1
1
10
4
7
1
4
11
1
10
4
7
1
4
7
7
1
13
68
2
44
27
23
2
3
3
2
22
3
29
2
27
2
22
2
19
2
3
19
3
2
23
3
2
2
19
12
3
2
19
12
2
22
3
2
3
69
2
3
2
12
2
19
26
2
19
Table 12. Rows 94-140; Columns 54-71
69
17
1
5
1
17
4
5
1
5
17
5
11
1
5
1
4
11
1
38
7
11
1
7
1
4
1
5
4
1
1
1
5
4
5
1
4
5
1
5
4
5
1
4
5
5
1
25
70
19
9
18
9
19
9
18
9
18
19
15
19
9
15
18
9
19
9
30
18
19
9
23
18
9
30
15
9
18
9
19
76
9
35
15
9
19
15
18
9
15
19
9
19
26
9
19