International Journal of Pure and Applied Mathematics
Volume 101 No. 6 2015, 1003-1011
ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)
url: http://www.ijpam.eu
AP
ijpam.eu
SKEW CHROMATIC INDEX OF COMB, LADDER,
AND MOBIUS LADDER GRAPHS
Joice Punitha M.1 , S. Rajakumari2
1 Department of Mathematics
L.N. Government College (Autonomous)
Ponneri, 601 204, Tamilnadu, INDIA
2 Department of Mathematics
R.M. D Engineering College
Kavaraipettai, 601 206, Tamilnadu, INDIA
Abstract: A skew edge coloring of a graph G is defined to be a set of two edge
colorings such that no two edges are assigned the same unordered pair of colors.
The skew chromatic index s(G) is the minimum number of colors required for a
skew edge coloring of G. In this paper, an algorithm is determined for skew edge
coloring of comb, ladder and Mobius ladder graphs. Also the skew chromatic
index of these graphs is solved in polynomial time.
AMS Subject Classification: 05C15
Key Words: skew edge coloring; skew chromatic index; comb graph; ladder
graph; Mobius ladder graph
1. Introduction
Let G = (V, E) be a simple, connected undirected graph with vertex set V and
edge set E. An edge coloring of a graph G is an assignment of colors to the edges
of G so that no two adjacent edges are assigned the same color. The minimum
number of colors required for an edge coloring of G is the edge chromatic number
denoted by χ′ (G). By Vizing’s theorem, ∆(G) ≤ χ′ (G) ≤ ∆(G) + 1, where
∆(G) is the maximum degree of vertices in G. Edge coloring problems are
well studied in both computer science and mathematics [5] and [8]. In cellular
Received:
March 12, 2015
c 2015 Academic Publications, Ltd.
url: www.acadpubl.eu
1004
J. Punitha M., S. Rajakumari
communication, frequency reusing is done by modeling it as an edge coloring
problem to avoid co-channel interference [11]. In this paper, we consider skew
edge coloring problems which are inspired from the study of skew Room squares
by R. A. Brualdi [4]. The concept of skew chromatic index was introduced by
Marsha F. Foregger and better upper bounds for s(G) was discussed when G
is cyclic, cubic or bipartite [6].
A skew edge coloring of G is an assignment of an ordered pair of colors
(ai , bi ) to each edge ei of G such that:
(i) the ai ’s form an edge coloring of G,
(ii) the bi ’s form an edge coloring of G, and
(iii)the pairs {ai , bi } are all distinct.
The two edge colorings are referred to as component colorings of the skew
edge coloring. The skew chromatic index s(G) is the minimum number of colors
required for a skew edge coloring of G.
In this paper, we have found an algorithm for skew edge coloring of comb,
ladder and Mobius ladder graphs and the skew chromatic index of these graphs
are solved in polynomial time. All the notations and definitions used in this
paper are as in [3].
2. Lower Bound on s(G)
Skew chromatic index, s(G) is the minimum number of colors used in two edge
colorings of G such that no two edges are assigned the same unordered pair
of colors. Each component coloring of a skew edge coloring is itself an edge
coloring. Therefore we have s(G) ≥ χ′ (G). Since ∆(G) ≤ χ′ (G) ≤
∆(G) + 1,
k+1
we have s(G) ≥ ∆(G). If k colors are used then there are
unordered
2
pairs of colors and this number must be at least as large asthe number
of edges
k+1
in G. Let k(m) denote the smallest integer k satisfying
≥ m where
2
m denotes the number of edges in G. Thus the best lower bound for s(G) is
s(G) ≥ max{∆(G), k(| E(G) |)} [6].
3. Comb Graph
Definition 1. A graph G = (V, E) is called a comb graph [1] if (i) it is
a tree (ii) all vertices are of degree at most three (iii) all the vertices of degree
SKEW CHROMATIC INDEX OF COMB, LADDER...
1005
three lie on a single simple path.
Let us consider a comb graph with 2n vertices and m = 2n − 1 edges. For
convenience, the edges are labeled as e1 , e2 , . . . , ek , ek+1 , . . . , em in a specific
manner as shown in Figure 1.
e2
e1
e3
e4
e5
ek
e6
...
ek-1
ek+1
ek+2
em-1
. ..e
m-2
em
Figure 1: Comb graph with 2n vertices and m edges
Algorithm 2. Algorithm for skew edge coloring of comb graphs for n ≥ 4.
Input: A comb graph with 2n vertices and m = 2n − 1 edges,
n ≥4.
k+1
Step 1: Find the smallest positive integer k such that
≥ m.
2
Step
2: Let {1, 2, 3, . . . , k} be the set of colors available to color the edges
k+1
with
unordered pairs of colors of the form {ai , bi } where ai ’s form the
2
first component coloring and bi ’s form the second component coloring.
Step 3: Skew edge coloring can be done as two component colorings.
First component coloring: Starting with edge e1 , ith edge in every set
of k edges of comb graph is colored using the ith color. When the number of
edges in the last set is less than k, its ith edge is colored using ith color till
all the edges of the graph are colored. This forms an edge coloring of the first
component. See Figure 2.
Second component coloring: Here, the 1st edge in j consecutive sets of k
edges is colored with the 1st , 2nd , 3rd , · · · , j th color respectively. The remaining
edges in each set are colored using the remaining colors continued from the
corresponding next color from the set {1, 2, 3, . . . , k} until all the colors are
exhausted. When the number of edges in the (j + 1)th set is less than k, then
its first edge is colored with (j + 1)th color and the subsequent edges are colored
using the subsequent colors from the set {1, 2, 3, . . . , k} taken in order. This
forms an edge coloring of the second component. See Figure 3.
Step 4: Form the two component colorings (ai , bi ) made of the first component coloring and the second component coloring of the ith edge. See Figure
4.
Output: Skew edge coloring of the comb graph.
1006
J. Punitha M., S. Rajakumari
2
2
4
3
1
5
3
1
6
4
6
5
2
3
1
6
4
5
2
3
1
Figure 2: First component coloring of comb graph with 2n = 22, m = 21
2
1
4
3
3
6
5
2
5
4
1
6
4
3
2
6
5
1
5
4
6
Figure 3: Second component coloring of comb graph with 2n = 22, m =
21
2, 2 4, 4 6, 6 2, 3
1, 1
4, 5 6, 1 2, 4 4, 6 6, 2
3,3 5, 5 1, 2 3, 4 5, 6 1, 3
2, 5
3,5 5, 1 1, 4 3, 6
Figure 4: Skew edge coloring of comb graph with 2n = 22, m = 21
SKEW CHROMATIC INDEX OF COMB, LADDER...
1007
k+1
Proof. Let e1 , e2 , e3 , . . . , em be the edges of G. Fix k such that
≥m
2
k+1
so that
unordered pairs are available for skew edge coloring. The
2
coloring of edges is done in a specific order as in Figure 1. First k edges
e1 , e2 , e3 , . . . , ek are assigned the colors in such a way that all ordered pairs of
the form (i, i), i = 1, 2, 3, . . . , k are used. In the second set of k edges, the first
k − 1 edges are assigned the colors of the form (i, i + 1), i = 1, 2, 3, . . . , k − 1.
As only k colors are considered, the ordered pair (k, k + 1) corresponding to
(i = k) is not permissible. The kth edge is assigned the color (k, 1) which can
be written as ((k − 1) + i, i), i = 1. In the next set of k edges, the first k − 2
edges are assigned colors of the form (i, i + 2), i = 1, 2, 3, . . . , k − 2. As the
ordered pairs (k − 1, k + 1) and (k, k + 2) corresponding to (i = k − 1, k) are
not permissible, the (k − 1)th edge and the kth edge are assigned the colors
((k − 2) + i, i), i = 1, 2 respectively. This process is continued till all the edges
are colored and the above method of edge coloring assigns only distinct ordered
pairs from the available k colors. See Figure 4.
Theorem 3. Let G be a comb
graphmwith 2n vertices and m = 2n − 1
l
√
−1+ 1+8m
.
edges, n ≥ 4. Then s(G) = k =
2
Proof. As there are m edges, at least m ordered pairs of colors are required
k+1
for skew edge coloring of G. If k colors are used, then there will be
2
k+1
pairs of colors. This
must be at least as large as the number of edges
2
k+1
in G. Therefore fix k in such a way that
≥ m. i.e. (k+1)k
≥ m .
2
2
It follows that k2 + k ≥ 2m. i.e.k 2 + k −√2m ≥ 0. Thus solving for k and
−1+ 1+8m
≥ 0, and its greatest integer
takingl the positive
2
m root, we obtain k =
k =
√
−1+ 1+8m
2
will be the minimum number of colors used in skew edge
m
l
√
coloring. Therefore s(G) = k = −1+ 21+8m .
Thus, based on the lower bound for s(G), we obtain an optimal solution for
skew chromatic index of comb graphs for n ≥ 4.
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J. Punitha M., S. Rajakumari
4. Ladder Graph
Definition 4. A ladder graph Ln [2] is a planar undirected graph with 2n
vertices and m = 3n − 2 edges where ∆(Ln ) = 3 and δ(Ln ) = 2.
For convenience, the edges are labeled as e1 , e2 , e3 , . . . , ek , ek+1 , . . . , em in a
specific manner as shown in Figure 5.
e2
e1
e5
e3
e4
ek
e8
e6
e7
ek+1
...
e10
e13
ek-1
ek+3
...
ek+2
em-2
em-1
em
Figure 5: Generalized ladder graph
Algorithm 5. Algorithm for skew edge coloring of ladder graphs, n ≥ 6.
Input: A ladder graph Ln with 2n vertices and m = 3n − 2 edges, n ≥ 6.
Two component colorings of a ladder graph can be obtained in a manner
similar to that of skew edge coloring of comb graph by following the steps 1 − 4
as in algorithm 2. See Figure 6.
Output: Skew edge coloring of the ladder graph.
2, 2
1, 1
5, 5 1, 2 4, 5 7, 1 3, 5 6, 1 2, 5 5, 1
3,3 6, 6 2, 3 5, 6 1, 3 4, 6
4, 4 7, 7 3, 4 6, 7 2, 4 5, 7
7,2 3, 6 6, 2
1, 4 4, 7
7, 3
Figure 6: Skew edge coloring of ladder graph with 2n = 20, m = 28 and
s(G) = 7
Theorem 6. Let Ln be a ladder
graph mwith 2n vertices and m = 3n − 2
l
√
−1+ 24n−15
.
edges, n ≥ 6. Then s(G) = k =
2
Thus, based on the lower bound for s(G), we prove that the skew chromatic
index for ladder graph Ln is tight for n ≥ 6.
SKEW CHROMATIC INDEX OF COMB, LADDER...
1009
5. Mobius Ladder Graph
A Mobius ladder graph Mn is a simple cubic graph on 2n vertices and 3n edges
[10]. Mobius ladder graph was introduced by Richard Guy and Frank Harary
in 1966 [7].
Definition 7. A Mobius ladder graph Mn [9] is a graph obtained from the
ladder Pn × P2 by joining the opposite end points of the two copies of Pn . For
convenience, the edges are labelled as e1 , e2 , e3 , . . . , ek , ek+1 , . . . , em−1 , em in a
specific manner as shown in Figure 7.
e2
e5
ek
e8
ek+3
em-4
em
e1
e3
e6
ek+1
...
...
em-3
e m-1
e4
e7
ek-1
ek+2
em-2
Figure 7: Generalized Mobius ladder graph
Algorithm 8. Algorithm for skew edge coloring of Mobius ladder graphs,
n ≥ 6.
Input: A Mobius ladder graph Mn with 2n vertices and m = 3n edges,
n ≥ 6.
Step 1: Consider the subgraph of Mobius ladder graph Mn without the
diagonal edges which is a ladder graph Ln with 2n vertices and 3n − 2 edges.
By proceeding as in algorithm 5, obtain the two component colorings (ai , bi ) of
the subgraph Ln for each of its edges ei , i = 1, 2, . . . , m − 2.
Step 2: Obtain the pairs of colors to color the edges em−1 and em based
on the coloring components of the edge em−2 .
Case 1: When both the coloring components of em−2 namely (am−2 , bm−2 )
are less than k−1, then form the pairs (am−2 +1, bm−2 +1), (am−2 +2, bm−2 +2).
The two pairs thus formed are distinct with each of the coloring components
being less than or equal to k.
Case 2: When atleast one of the coloring components of the edge em−2
is either k or k − 1, then consider the coloring of the edges em−2 , em−3 , em−4
viz. (am−2 , bm−2 ), (am−3 , bm−3 ), (am−4 , bm−4 ) and form the pairs (bm−2 , am−3 ),
(bm−3 , am−4 ) and if possible, obtain the ordered pairs (bm−2 +1, am−3 +1), (bm−3 +
1, am−4 + 1) provided each of the coloring components is less than or equal to
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J. Punitha M., S. Rajakumari
k. Out of these four pairs, two or three are distinct.
Step 3: Assign colors to the edges em−1 and em based on the components
of the pairs obtained in step 2.
Case 1: If one of the coloring components is 1, do not use that pair for
coloring the edges.
Case 2: If the coloring components are independent of the colors 2 or 4,
then assign those pairs to the edges em−1 and em taken in order.
Case 3: If one of the coloring components is 2, then assign it to the edge
em−1 and the next pair to the edge em .
Case 4: If one of the coloring components is 4, then assign it to the edge
em and the next pair to the edge em−1 .
Output: Skew edge coloring of the Mobius ladder graph. See Figure 8.
5, 5 8, 8 3, 4 6, 7 1, 3 4, 6 7, 1 2 5
7, 3
3,3 6, 6 1, 2 4, 5 7, 8 2, 4 5,7 8, 2 3, 6
2, 2
1, 1
6, 2
4, 4 7, 7 2, 3 5, 6 8, 1 3, 5
6, 8 1, 4
4, 7
Figure 8: Skew edge coloring of Mobius ladder graph with 2n = 20, m =
30 and s(G) = 8
Theorem 9. Let Mn be a Mobius
ladder
m graph with 2n vertices and
l
√
−1+ 1+24n
.
m = 3n edges, n ≥ 6. Then s(G) =
2
Thus, based on the lower bound for s(G), we obtain an optimal solution for
skew chromatic index of Mobius ladder graphs for n ≥ 6.
6. Conclusion
In this paper, we have designed algorithms for skew edge coloring of comb,
ladder and Mobius ladder graphs and obtained an optimal solution for s(G). It
would be interesting to identify the skew chromatic index for various networks.
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