Cent. Eur. J. Math. • 12(12) • 2014 • 1882-1889
DOI: 10.2478/s11533-014-0449-3
Central European Journal of Mathematics
A maximum degree theorem
for diameter-2-critical graphs
Research Article
Teresa W. Haynes1,2∗ , Michael A. Henning2† , Lucas C. van der Merwe3‡ , Anders Yeo2,4§
1 Department of Mathematics and Statistics, East Tennessee State University, Johnson City, TN 37614-0002, USA
2 Department of Mathematics, University of Johannesburg, Auckland Park, South Africa
3 Department of Mathematics, University of Tennessee in Chattanooga, Chattanooga, TN 37403, USA
4 Engineering Systems and Design, Singapore University of Technology and Design, 20 Dover Drive Singapore, 138682,
Singapore
Received 26 July 2013; accepted 10 March 2014
Abstract: A graph is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. Let G
be a diameter-2-critical graph of order n. Murty and Simon conjectured that the number of edges in G is at most
bn2 /4c and that the extremal graphs are the complete bipartite graphs Kbn/2c,dn/2e . Fan [Discrete Math. 67 (1987),
235–240] proved the conjecture for n ≤ 24 and for n = 26, while Füredi [J. Graph Theory 16 (1992), 81–98]
proved the conjecture for n > n0 where n0 is a tower of 2’s of height about 1014 . The conjecture has yet to be
proven for other values of n. Let ∆ denote the maximum degree of G. We prove the following maximum degree
theorems for diameter-2-critical graphs. If ∆ ≥ 0.7 n, then the Murty-Simon Conjecture is true. If n ≥ 2000 and
∆ ≥ 0.6789 n, then the Murty-Simon Conjecture is true.
MSC:
05C65
Keywords: Diameter critical • Diameter-2-critical • Total domination critical
© Versita Sp. z o.o.
∗
†
‡
§
1882
E-mail:
E-mail:
E-mail:
E-mail:
[email protected]
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[email protected]
T.W. Haynes et al.
1.
Introduction
Distance and diameter are fundamental concepts in graph theory. A graph G is called diameter-2-critical if its diameter
is two, and the deletion of any edge increases the diameter. Diameter-2-critical graphs are extensively studied in the
literature. See, for example, [1–3, 5, 6, 10–12, 16, 17] and elsewhere.
Plesník [17] observed that all known diameter-2-critical graphs on n vertices have no more than n2 /4 edges and that the
extremal graphs appear to be the complete bipartite graphs with partite sets whose cardinality differs by at most one.
Murty and Simon (see [2]) independently made the following conjecture:
Murty-Simon Conjecture.
If G is a diameter-2-critical graph with n vertices and m edges, then m ≤ bn2 /4c, with equality if and only if G is the
complete bipartite graph Kb n c,d n e .
2
2
According to Füredi [6], Erdős said that this conjecture goes back to the work of Ore in the 1960s. Although considerable
work has been done in an attempt to completely resolve the conjecture (see, for example, [2, 5, 6, 9–12, 17] and elsewhere)
and several impressive partial results have been obtained, the conjecture remains open for general n. Fan [5] proved the
conjecture for n ≤ 24 and for n = 26. In 1992 Füredi [6] gave an asymptotic result proving the conjecture is true for
large n, that is, for n > n0 where n0 is a tower of 2’s of height about 1014 . The conjecture has yet to be proven for other
values of n.
2.
Main result
Our aim in this paper is to prove the Murty-Simon Conjecture for graphs with sufficiently large maximum degree. More
precisely, we shall prove the following:
Theorem 2.1.
Let G be a diameter-2-critical of order n and size m. Let ∆ = ∆(G). Then the following holds.
(a) If ∆ ≥ 0.7 n, then m < bn2 /4c.
(b) If n ≥ 2000 and ∆ ≥ 0.6789 n, then m < bn2 /4c.
3.
Proof of Theorem 2.1
Before proving Theorem 2.1 in Section 3.4, we give some terminology and preliminary results.
3.1.
Graph theory terminology
For notation and graph theory terminology, we in general follow [8]. We denote the complement of a graph G by G.
A set S of vertices in G is a dominating set in G if every vertex of V (G) \ S is adjacent to a vertex in S. A total
dominating set in G is a set S of vertices of G such that every vertex is adjacent to a vertex in S. The total domination
number γt (G) is the minimum cardinality of a total dominating set in G. Total domination is now very well studied in
graph theory. For more details, the reader is referred to the book on domination theory by Haynes, Hedetniemi, and
Slater [8] and a recent survey on total domination [14].
Let G = (V , E) be a graph. If S ⊆ V , and u and v are two nonadjacent vertices that belong to S, then we say that uv
is a missing edge in S (rather than “uv is a missing edge in the induced subgraph G[S]”). Also, if there are no missing
edges in S, we take the liberty to write that S is complete (rather than “G[S] is complete”).
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A maximum degree theorem for diameter-2-critical graphs
For disjoint subsets X and Y of vertices in a graph G, we denote by [X , Y ] the set of all edges in G that join vertices
in X and vertices in Y . For a vertex v ∈ V and a subset X ⊆ V , the X -neighborhood of v is the set NX (v) = NG (v) ∩ X
and the X -closed neighborhood of v is the set NX [v] = NX (v) ∪ {v}. Thus, NX (v) is the set of neighbors of v that belong
to the set X . Further, the degree of v in X is given by dX (v) = |NX (v)|; that is, dX (v) is the number of vertices in X that
are adjacent to v.
3.2.
The association with total domination
In order to prove our main result, we use an important association of diameter-2-critical graphs with total domination
in graphs. Conventionally the diameter of a disconnected graph is considered to be either undefined or defined as
infinity. For the purposes of this paper, we use the former and hence require that a diameter-2-critical graph is 2-edge
connected. This implies that the graphs we consider will have minimum degree two (since removing an edge incident to
a vertex of degree one results in a disconnected graph). We note, however, that if we relaxed the condition and defined
the diameter of disconnected graphs to be infinity, the only additional diameter-2-critical graphs are stars with order
at least three, that is, the complete bipartite graphs with exactly one partite set of cardinality 1.
As introduced in [13], a non-complete graph G is total domination edge critical if γt (G + e) < γt (G) for every edge
e ∈ E(G). Further if γt (G) = k, then we say that G is a kt -critical graph. Thus if G is kt -critical, then its total
domination number is k and the addition of any edge decreases the total domination number. It is shown in [13] that the
addition of an edge to a graph can change the total domination number by at most two. Total domination edge critical
graphs G with the property that γt (G) = k and γt (G + e) = k − 2 for every edge e ∈ E(G) are called kt -supercritical
graphs.
Hanson and Wang [7] were the first to observe the following intermediary relationship between diameter-2-critical graphs
and total domination edge critical graphs. Note that this relationship is contingent on total domination being defined
in the complement G of the diameter-2-critical graph G, that is, G has no isolated vertices. Hence our elimination of
stars as diameter-2-graphs is necessary for this association.
Theorem 3.1 ([7]).
A graph is diameter-2-critical if and only if its complement is 3t -critical or 4t -supercritical.
The 4t -supercritical graphs are characterized in [20].
Theorem 3.2 ([20]).
A graph G is 4t -supercritical if and only if G is the disjoint union of two nontrivial complete graphs.
Bounds on the diameter of 3t -critical graphs were established in [13].
Theorem 3.3 ([13]).
If G is a 3t -critical graph, then 2 ≤ diam(G) ≤ 3.
3.3.
Quasi-cliques
Since γt (G) ≥ 2 for any graph G, the addition of an edge to a 3t -critical graph reduces the total domination number by
exactly one. Hence if G is a 3t -critical graph, then γt (G) = 3 and γt (G + e) = 2 for every edge e ∈ E(G) 6= ∅. We will
frequently use the following observation and notation.
Observation 3.4 ([10]).
For every 3t -critical graph G and nonadjacent vertices u and v in G, either the set {u, v} is a dominating set in G or,
without loss of generality, the set {u, w} is a dominating set in G − v for some vertex w adjacent to u but not to v in
G. In this case, we write uw 7→ v.
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T.W. Haynes et al.
Let G = (V , E) be a 3t -critical graph, and let uv be a missing edge such that {u, v} does not dominate G. By
Observation 3.4, uw 7→ v or vw 7→ u for some vertex w ∈ V \ {u, v}. If uw 7→ v, then in G + uv we have that {u, w}
dominates V . Further, v is the unique vertex not dominated by {u, w} in G. Analogously, if vw 7→ u, then in G + uv
we have that {v, w} dominates V and that u is the unique vertex not dominated by {v, w} in G. Hence there exists
an edge xy ∈ E(G + uv) such that in G + uv, {x, y} dominates V . The edge xy may not be unique. However, we
select one such edge xy and call it the quasi-edge for uv, abbreviated q.e.. (Thus for each edge in E(G), we associate a
unique quasi-edge.) For the purpose of counting edges, we say that a set S of k vertices forms a quasi-clique if we can
uniquely associate a quasi-edge for every non-edge in S. The number of such quasi-edges is k2 − |E(G[S])|. Further if
S is a quasi-clique, and e is an edge of G associated with a missing edge in S, then we call e a quasi-edge associated
with the quasi-clique S.
3.4.
Proof of Theorem 2.1
As a special case of a classic result of Turán [19] in 1941, we have the following bound on the maximum number of edges
in a triangle-free graph due to Mantel [15].
Mantel’s Theorem.
If G is a triangle-free graph with n vertices and m edges, then m ≤ bn2 /4c, with equality if and only if G is the complete
bipartite graph Kb n c,d n e .
2
2
As an immediate consequence of Mantel’s Theorem, we observe that the Murty-Simon Conjecture is true for triangle-free
graphs. Hanson and Wang [7] were the first to observe that the complement of a 4t -supercritical graph is a complete
bipartite graph. The number of edges in a complete bipartite graph is maximized when the partite sets differ in cardinality
by at most one, and so the Murty-Simon Conjecture holds for this case and a subset of the complements of 4t -supercritical
graphs yield the extremal graphs of the conjecture. In order to prove the Murty-Simon Conjecture, it suffices to prove
that if G is a diameter-2-critical graph with n vertices and m edges and if G is not a complete bipartite graph, then
m < bn2 /4c. By Theorem 3.2, the complement G of such a graph is not a 4t -supercritical
graph and therefore,
by
j k
l
m
Theorem 3.1, is a 3t -critical graph. We note that for a graph G of order n, |E(G)| < n4 if and only if |E(G)| > n(n−2)
.
4
Therefore, by Theorem 3.1 and Theorem 3.2, the Murty-Simon Conjecture is equivalent to the following conjecture.
2
Conjecture 3.5.
If G is a 3t -critical graph with order n and size m, then m >
l
n(n−2)
4
m
.
In order to prove our main theorem, namely Theorem 2.1, it therefore suffices for us to prove the following theorem about
total domination critical graphs, as δ(G) = n − 1 − ∆(G).
Theorem 3.6.
Let G be a 3t -critical graph of order n and size m. Let δ = δ(G) ≥ 1. Then the following holds.
(a) If δ ≤ 0.3 n, then m > dn(n − 2)/4e.
(b) If n ≥ 2000 and δ ≤ 0.321 n, then m > dn(n − 2)/4e.
Proof.
As observed earlier, Fan [5] proved the Murty-Simon Conjecture for n ≤ 24 and for n = 26. Hence for these
values of n, we note that Conjecture 3.5 is true; that is, if n ≤ 24 or n = 26, then m > dn(n − 2)/4e and Part (a) holds.
Hence in what follows we restrict our attention to n ≥ 25.
Recall that by Theorem 3.3, we have that 2 ≤ diam(G) ≤ 3. It is proven in [10] that Conjecture 3.5 is true for graphs G
with diam(G) = 3. Hence in what follows we may assume that diam(G) = 2, for otherwise there is nothing left to prove.
Let δ = δ(G), where by assumption δ ≥ 1, and let v be a vertex in G of degree δ. Let A = N(v) and B = V \ N[v].
Since diam(G) = 2, we note that every vertex in B is adjacent to at least one vertex in A. Let x be a vertex in A whose
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A maximum degree theorem for diameter-2-critical graphs
degree in G[A] is a minimum, and let dA (x) = k. Further, let X = A \ NA [x], and so X is the set of all vertices in A that
are not adjacent to x in G. Since |A| = δ, we note that |X | = δ − k − 1 and |B| = n − δ − 1. We proceed further with
the following series of claims.
Claim A.
The set B forms a quasi-clique.
Proof of Claim A. If B is a clique in G, then B is a quasi-clique, as desired. Hence we may assume that B is not a
clique. Let uw be a missing edge in B. We note that the vertex v is not dominated by {u, w}. Hence by Observation 3.4
we may assume, renaming u and w if necessary, that uz 7→ w for some vertex z. In order to dominate the vertex v, we
have that z ∈ A, and so uz ∈ [A, B]. Every vertex different from w is adjacent to u or z, and so w is the only vertex that
is adjacent to neither u nor z. We may therefore uniquely associate the edge uz with the missing edge uw. Hence, B
forms a quasi-clique.
Claim B.
The set X forms a quasi-clique.
Proof of Claim B. If X is a clique in G, then X is a quasi-clique, as desired. Hence we may assume that X is not a
clique. Let uw be a missing edge in X . We note that the vertex x is not dominated by {u, w}. Hence by Observation 3.4
we may assume, renaming u and w if necessary, that uz 7→ w for some vertex z. Since x is not adjacent to any vertex
of X , we note that z ∈
/ X ∪ {x}. Further since vw is an edge of G, we note that z 6= v. Every vertex different from w is
adjacent to u or z, and so w is the only vertex that is adjacent to neither u nor z. We may therefore uniquely associate
the edge uz with the missing edge uw. Hence, X forms a quasi-clique.
By Claim A and Claim B and their proofs, we note that no edge incident with v in G is associated with a missing edge
in the quasi-clique B or the quasi-clique X . Neither is an edge incident with x in G[A] associated with a missing edge
in the quasi-clique B or the quasi-clique X . Furthermore no edge in G[A] is associated with a missing edge in the
quasi-clique B. Hence letting f(k) denote the number of edges in G that are neither incident with the vertex v nor
associated with the quasi-clique B, we have that
|X |
f(k) ≥ dA (x) +
2
and
f(k) ≥ m(G[A]).
Hence since dA (x) = k, |X | = δ − k − 1, and m(G[A]) ≥ 12 |A|k = 21 δk, we have that
f(k) ≥ max
1 2
1
2
(δ − 2δk − 3δ + 5k + k + 2), δk .
2
2
In order to minimize the function f(k), treating δ as a constant, we note that the function 12 (δ 2 − 2δk − 3δ + 5k + k 2 + 2)
is monotone decreasing on the interval [0, δ], while the function 12 δk is monotone increasing on the interval [0, δ]. Let
θ = δ(δ−3)/2+1 and note that this is 21 (δ 2 −2δk −3δ+5k +k 2 +2) evaluated at k = 0. As 0 ≤ θ ≤ δ 2 /2, where 0 and δ 2 /2
is δk/2 evaluated at k = 0 and k = δ, respectively, we note that 21 (δ 2 − 2δk − 3δ + 5k + k 2 + 2) and 12 δk intersect in [0, δ].
Therefore the function f(k) achieves its minimum value when k satisfies the equation 21 (δ 2 −2δk −3δ +5k +k 2 +2) = 12 δk.
Solving this equation yields
√
−5 + 3δ − 17 − 18δ + 5δ 2
k=
2
(recall that k < δ). This implies that
f(k) ≥
1886
δ(−5 + 3δ −
√
17 − 18δ + 5δ 2 )
.
4
(1)
T.W. Haynes et al.
Since the size of G is at least
m(G) ≥ dG (v) + f(k) +
|B|
2
and dG (v) = δ and |B| = n − δ − 1, we have by Equation (1) that
m(G) ≥
√
1 2
2n − 4δn − 6n + 4 + 5δ 2 + 5δ − δ 17 − 18δ + 5δ 2 .
4
(2)
If n is even, we wish to show that
m(G) −
n(n − 2)
n(n − 2)
= m(G) −
> 0.
4
4
Hence, by Equation (2), for n even it suffices for us to consider the function
geven (n, δ) =
√
1 2
n − 4δn − 4n + 4 + 5δ 2 + 5δ − δ 17 − 18δ + 5δ 2
4
and to show that geven (n, δ) > 0 (since m(G) − n(n − 2)/4 ≥ geven (n, δ) by Equation (2)). If n is odd, we wish to show
that
n(n − 2)
(n − 1)2
m(G) −
= m(G) −
> 0.
4
4
Hence, by Equation (2), it suffices for us to consider the function
godd (n, δ) =
√
1 2
n − 4δn − 4n + 3 + 5δ 2 + 5δ − δ 17 − 18δ + 5δ 2
4
and to show that godd (n, δ) > 0 (since m(G) − (n − 1)2 /4 ≥ godd (n, δ) by Equation (2)).
For all n let g(n, δ) = godd (n, δ) and note that geven (n, δ) ≥ godd (n, δ) = g(n, δ) for all n. Therefore it suffices to show
that g(n, δ) > 0 for all n. Let c be defined such that δ = cn. We now prove the following claim.
Claim C. √
If n(c2 (5 −
5) − 4c + 1) > 4 − 5c, then g(n, δ) > 0.
Proof of Claim C. Assume that n(c2 (5 −
proof of Claim C.
√
5) − 4c + 1) > 4 − 5c. The following holds, as δ ≥ 1, which completes the
n(c2 (5 −
√
5) − 4c + 1) − (4 − 5c) > 0
m
√
n[n(5c2 − 4c + 1) + (5c − 4 − c2 n 5)] > 0
m
√
5δ 2 − 4δn + n2 + 5δ − 4n − δ 2 5 > 0
⇓ √
5δ 2 − 4δn + n2 + 5δ − 4n − δ 5δ 2 + 17 − 18δ + 3 > 0
⇓
g(n, δ) > 0.
√
By Claim
C we note that if c2 (5 − 5) − 4c + 1 > 0, then g(n, δ) > 0 for all sufficiently large n. Further we note that
√
c2 (5 − 5) − 4c + 1 = 0 when c satisfies the following.
1887
A maximum degree theorem for diameter-2-critical graphs
c=
4±
q
√
p√
16 − 4(5 − 5)
2±
5−1
√
√
=
.
2(5 − 5)
(5 − 5)
The above roots are approximately 0.321358865655 and 1.125854729845, implying that if c < 0.32135886565, then
g(n, δ) > 0 for all sufficiently large n when δ = cn. To prove Part (a), we consider the function
h(c) =
c2 (5 −
4 − 5c
√
,
5) − 4c + 1
where 0 < c < 1.
Claim D.
If c ≤ 0.3, then 51.28 > h(0.3) ≥ h(c).
Proof of Claim D. The desired result follows readily from the observation that h(c) is an increasing function of c in the
interval (0, 0.3] and the fact that h(0.3) is approximately 51.2779679.
If n > h(c), then by Claim C, we have g(n, δ) > 0. Hence by Claim D, if c ≤ 0.3 and n ≥ 52, then g(n, δ) > 0. In fact
given any n ≥ 24, Table 1 in Appendix shows the values of δ and c that will cause g(n, δ) > 0 to be true. In particular,
we note that if c ≤ 0.3 and 24 ≤ n < 52, we also have g(n, δ) > 0 (by Table 1). This proves Part (a) of the theorem.
Part (b) can be proven analogously (see Table 1 for values of δ and c that will cause g(n, δ) > 0 to be true).
Appendix
Table 1.
Values of n, δ and c such that g(n, δ) > 0 where δ = cn.
n
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
1888
δ
7.5186637
7.8409722
8.1631968
8.4853483
8.8074355
9.1294661
9.4514464
9.7733820
10.0952774
10.4171368
10.7389637
11.0607612
11.3825319
11.7042783
12.0260024
12.3477062
12.6693913
12.9910591
13.3127110
13.6343483
13.9559719
14.2775829
14.5991821
14.9207704
15.2423484
15.5639170
15.8854765
16.2070277
16.5285711
c
0.3007465
0.3015759
0.3023406
0.3030482
0.3037047
0.3043155
0.3048854
0.3054182
0.3059175
0.3063864
0.3068275
0.3072434
0.3076360
0.3080073
0.3083590
0.3086927
0.3090095
0.3093109
0.3095979
0.3098716
0.3101327
0.3103822
0.3106209
0.3108494
0.3110683
0.3112783
0.3114799
0.3116736
0.3118598
n
54
55
56
57
58
59
60
70
80
90
100
200
300
400
500
600
700
800
900
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
δ
16.8501071
17.1716361
17.4931585
17.8146748
18.1361852
18.4576902
18.7791899
21.9939521
25.2084055
28.4226579
31.6367720
63.7749649
95.9115998
128.0478569
160.1839646
192.3199982
224.4559895
256.5919545
288.7279019
320.8638372
642.2229214
963.5818598
1284.9407617
1606.2996491
1927.6585293
2249.0174053
2570.3762787
2891.7351504
3213.0940209
c
0.3120390
0.3122116
0.3123778
0.3125382
0.3126928
0.3128422
0.3129865
0.3141993
0.3151051
0.3158073
0.3163677
0.3188748
0.3197053
0.3201196
0.3203679
0.3205333
0.3206514
0.3207399
0.3208088
0.3208638
0.3211115
0.3211940
0.3212352
0.3212599
0.3212764
0.3212882
0.3212970
0.3213039
0.3213094
T.W. Haynes et al.
Acknowledgements
Second author is supported in part by the South African National Research Foundation and the University of Johannesburg.
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