Induced matching extendable graphs

We say that a simple graph G is induced matching extendable, shortly IM-extendable, if every induced matching of G is included in a perfect matching of G. The main results of this paper are as follows: (1) For every connected IM-extendable graph G with |V(G)| ≥ 4, the girth g(G) ≤ 4. (2) If G is a connected IM-extendable graph, then |E(G)| ≥ ${3\over 2}|V(G)| - 2$; the equality holds if and only if G ≅ T × K₂, where T is a tree. (3) The only 3-regular connected IM-extendable graphs are C_n × K₂, for n ≥ 3, and C_2n(1, n), for n ≥ 2, where C_2n(1, n) is the graph with 2n vertices x₀, x₁, …, x_2n−1, such that x_ix_j is an edge of C_2n(1, n) if either |i − j| ≡ 1 (mod 2n) or |i − j| ≡ n (mod 2n). © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 203–213, 1998     

The symmetric (2k,k)‐graphs

A noncomplete graph G is called an (n, k)‐graph if it is n‐connected and G − X is not (n − |X| + 1)‐connected for any X ⊆ V(G) with |X| ≤ k. Mader conjectured that for k ≥ 3 the graph K_{2k + 2} − (1‐factor) is the unique (2k, k)‐graph. We settle this conjecture for strongly regular graphs, for edge transitive graphs, and for vertex transitive graphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 36: 35–51, 2001     

The size of graphs without nowhere-zero 4-flows

Let G be a 2-edge-connected simple graph with order n. We show that if | V(G)| ≤ 17, then either G has a nowhere-zero 4-flow, or G is contractible to the Petersen graph. We also show that for n large, if | V(G)| n ? 17/2 + 34, then either G has a nonwhere-zero 4-flow, or G can be contracted to the Petersen graph. © 1995 John Wiley & Sons, Inc.     

Distinguishing Cartesian powers of graphs

The distinguishing number D(G) of a graph is the least integer d such that there is a d‐labeling of the vertices of G that is not preserved by any nontrivial automorphism of G. We show that the distinguishing number of the square and higher powers of a connected graph G ≠ K₂, K₃ with respect to the Cartesian product is 2. This result strengthens results of Albertson [Electron J Combin, 12 ( 1 ), #N17] on powers of prime graphs, and results of Klav?ar and Zhu [Eu J Combin, to appear]. More generally, we also prove that d(G □ H) = 2 if G and H are relatively prime and |H| ≤ |G| < 2^|H| ? |H|. Under additional conditions similar results hold for powers of graphs with respect to the strong and the direct product. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 250–260, 2006     

On the toughness index of planar graphs

The toughness indexτ(G) of a graph G is defined to be the largest integer t such that for any S ? V(G) with |S| > t, c(G - S) < |S| - t, where c(G - S) denotes the number of components of G - S. In particular, 1-tough graphs are exactly those graphs for which τ(G) ≥ 0. In this paper, it is shown that if G is a planar graph, then τ(G) ≥ 2 if and only if G is 4-connected. This result suggests that there may be a polynomial-time algorithm for determining whether a planar graph is 1-tough, even though the problem for general graphs is NP-hard. The result can be restated as follows: a planar graph is 4-connected if and only if it remains 1-tough whenever two vertices are removed. Hence it establishes a weakened version of a conjecture, due to M. D. Plummer, that removing 2 vertices from a 4-connected planar graph yields a Hamiltonian graph.     

Hamiltonian weights and unique 3-edge-colorings of cubic graphs

A (1,2)-eulerian weight w of a grph is hamiltonian if every faithful cover of w is a set of two Hamilton circuits. Let G be a 3-connected cubic graph containing no subdivition of the Petersen graph. We prove that if G admits a hamiltonian weight then G is uniquely 3-edge-colorable. © 1996 John Wiley & Sons, Inc.     

On a conjecture of wang and williams

Wang and Williams defined a threshold assignment for a graph G as an assignment of a non-negative weight to each vertex and edge of G, and a threshold t, such that a set S of vertices is stable if and only if the total weight of the subgraph induced by S does not exceed t - 1. This is always possible with zero vertex-weights, unit edge-weights, and t = 1. By definition, a threshold graph is a graph having a threshold assignment with zero edge-weights. The threshold weight of G is the minimum total edgeweight of a threshold assignment of G. A graph G = (V,E) is called heavy if its threshold weight is |E|. A clique C of G is called big if |E(C)| > |E(G - C)|. Wang and Williams showed that graphs with a big clique are not heavy, and conjectured the converse. They verified the conjecture for triangle-free graphs. We disprove the conjecture in general, but prove it for perfect graphs.     

A Construction of Uniquely C4-free colourable Graphs

Abstract

An F-free colouring of a graph G is a partition {V₁,V₂,…,V_n} of the vertex set V(G) of G such that F is not an induced subgraph of G[V_i] for each i. A graph is uniquely F-free colourable if any two .F-free colourings induce the same partition of V(G). We give a constructive proof that uniquely C₄-free colourable graphs exist.     

Constraints on the number of maximal independent sets in graphs

A maximal independent set of a graph G is an independent set that is not contained properly in any other independent set of G. Let i(G) denote the number of maximal independent sets of G. Here, we prove two conjectures, suggested by P. Erdös, that the maximum number of maximal independent sets among all graphs of order n in a family Φ is o(3^n/3) if Φ is either a family of connected graphs such that the largest value of maximum degrees among all graphs of order n in Φ is o(n) or a family of graphs such that the approaches infinity as n → ∞.     

Packing two forests into a bipartite graph

For two integers a and b, we say that a bipartite graph G admits an (a, b)-bipartition if G has a bipartition (X, Y) such that |X| = a and |Y| = b. We say that two bipartite graphs G and H are compatible if, for some integers a and b, both G and H admit (a, b)-bipartitions. In this note, we prove that any two compatible trees of order n can be packed into a complete bipartite graph of order at most n + 1. We also provide a family of infinitely many pairs of compatible trees which cannot be packed into a complete bipartite graph of the same order. A theorem about packing two forests into a complete bipartite graph is derived from this result. © 1996 John Wiley & Sons, Inc.     

Matching graphs

The matching graph M(G) of a graph G is that graph whose vertices are the maximum matchings in G and where two vertices M₁ and M₂ of M(G) are adjacent if and only if |M₁ − M₂| = 1. When M(G) is connected, this graph models a metric space whose metric is defined on the set of maximum matchings in G. Which graphs are matching graphs of some graph is not known in general. We determine several forbidden induced subgraphs of matching graphs and add even cycles to the list of known matching graphs. In another direction, we study the behavior of sequences of iterated matching graphs. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 73–86, 1998     

On minors of graphs with at least 3n −4 edges

A point disconnecting set S of a graph G is a nontrivial m-separator, where m = |S|, if the connected components of G - S can be partitioned into two subgraphs, each of which has at least two points. A 3-connected graph is quasi 4-connected if it has no nontrivial 3-separators. Suppose G is a graph having n ≥ 6 points. We prove three results: (1) If G is quasi 4-connected with at least 3n - 4 edges, then the graph K^?₁, obtained from K₆ by deleting an edge, is a minor of G. (2) If G has at least 3n - 4 edges then either K^?₆ or the graph obtained by pasting two disjoint copies of K₅ together along a triangle is a minor of G. (3) If the minimum degree of G is at least 6, then K^?₆ is a minor of G. © 1993 John Wiley & Sons, Inc.     

Total domination in 2‐connected graphs and in graphs with no induced 6‐cycles

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ_t(G) of G. It is known [J Graph Theory 35 (2000), 21–45] that if G is a connected graph of order n > 10 with minimum degree at least 2, then γ_t(G) ≤ 4n/7 and the (infinite family of) graphs of large order that achieve equality in this bound are characterized. In this article, we improve this upper bound of 4n/7 for 2‐connected graphs, as well as for connected graphs with no induced 6‐cycle. We prove that if G is a 2‐connected graph of order n > 18, then γ_t(G) ≤ 6n/11. Our proof is an interplay between graph theory and transversals in hypergraphs. We also prove that if G is a connected graph of order n > 18 with minimum degree at least 2 and no induced 6‐cycle, then γ_t(G) ≤ 6n/11. Both bounds are shown to be sharp. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 55–79, 2009     

P4‐decompositions of regular graphs

In this article, we show that every simple r‐regular graph G admits a balanced P₄‐decomposition if r ≡ 0(mod 3) and G has no cut‐edge when r is odd. We also show that a connected 4‐regular graph G admits a P₄‐decomposition if and only if |E(G)| ≡ 0(mod 3) by characterizing graphs of maximum degree 4 that admit a triangle‐free Eulerian tour. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 135–143, 1999     

A Canonical Ramsey Theorem

Say a graph H selects a graph G if given any coloring of H, there will be a monochromatic induced copy of G in H or a completely multicolored copy of G in H. Denote by s(G) the minimum order of a graph that selects G and set s(n) = max {s(G): |G| = n}. Upper and lower bounds are given for this function. Also, consider the Folkman function f_r(n) = max{min{|V(H)|: H → (G)¹_r}: |V(G)| = n}, where H → (G)¹_r indicates that H is vertex Ramsey to G, that is, any vertex coloring of H with r colors admits a monochromatic induced copy of G. The method used provides a better upper bound for this function than was previously known. As a tool, we establish a theorem for projective planes.     

Pan-H-Linked Graphs

Let H be a multigraph, possibly containing loops. An H-subdivision is any simple graph obtained by replacing the edges of H with paths of arbitrary length. Let H be an arbitrary multigraph of order k, size m, n ₀(H) isolated vertices and n ₁(H) vertices of degree one. In Gould and Whalen (Graphs Comb. 23:165–182, 2007) it was shown that if G is a simple graph of order n containing an H-subdivision H{\mathcal{H}} and d(G) 3 \fracn+m-k+n₁(H)+2n₀(H)2{\delta(G) \ge \frac{n+m-k+n_1(H)+2n_0(H)}{2}}, then G contains a spanning H-subdivision with the same ground set as H{\mathcal{H}} . As a corollary to this result, the authors were able to obtain Dirac’s famed theorem on hamiltonian graphs; namely that if G is a graph of order n ≥ 3 with d(G) 3 \fracn2{\delta(G)\ge\frac{n}{2}} , then G is hamiltonian. Bondy (J. Comb. Theory Ser. B 11:80–84, 1971) extended Dirac’s theorem by showing that if G satisfied the condition d(G) 3 \fracn2{\delta(G) \ge \frac{n}{2}} then G was either pancyclic or a complete bipartite graph. In this paper, we extend the result from Gould and Whalen (Graphs Comb. 23:165–182, 2007) in a similar manner. An H-subdivision H{\mathcal{H}} in G is 1-extendible if there exists an H-subdivision H^*{\mathcal{H}^{*}} with the same ground set as H{\mathcal{H}} and |H^*| = |H| + 1{|\mathcal{H}^{*}| = |\mathcal{H}| + 1} . If every H-subdivision in G is 1-extendible, then G is pan-H-linked. We demonstrate that if H is sufficiently dense and G is a graph of large enough order n such that d(G) 3 \fracn+m-k+n₁(H)+2n₀(H)2{\delta(G) \ge \frac{n+m-k+n_1(H)+2n_0(H)}{2}} , then G is pan-H-linked. This result is sharp.     

Existential closure of block intersection graphs of infinite designs having infinite block size

A graph G is n ‐existentially closed ( n ‐e.c.) if for each pair ( A, B ) of disjoint subsets of V(G) with | A | + | B |≤ n there exists a vertex in V ( G )\( A ∪ B ) which is adjacent to each vertex in A and to no vertex in B . In this paper we study the n ‐existential closure property of block intersection graphs of infinite designs with infinite block size. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:317‐327, 2011     

k-ordered Hamiltonian graphs

A hamiltonian graph G of order n is k-ordered, 2 ≤ k ≤ n, if for every sequence v₁, v₂, …, v_k of k distinct vertices of G, there exists a hamiltonian cycle that encounters v₁, v₂, …, v_k in this order. Theorems by Dirac and Ore, presenting sufficient conditions for a graph to be hamiltonian, are generalized to k-ordered hamiltonian graphs. The existence of k-ordered graphs with small maximum degree is investigated; in particular, a family of 4-regular 4-ordered graphs is described. A graph G of order n ≥ 3 is k-hamiltonian-connected, 2 ≤ k ≤ n, if for every sequence v₁, v₂, …, v_k of k distinct vertices, G contains a v₁-v_k hamiltonian path that encounters v₁, v₂,…, v_k in this order. It is shown that for k ≥ 3, every (k + 1)-hamiltonian-connected graph is k-ordered and a result of Ore on hamiltonian-connected graphs is generalized to k-hamiltonian-connected graphs. © 1997 John Wiley & Sons, Inc.     

Domination in graphs with minimum degree two

The domination number γ(G) of a graph G = (V, E) is the minimum cardinality of a subset of V such that every vertex is either in the set or is adjacent to some vertex in the set. We show that if a connected graph G has minimum degree two and is not one of seven exceptional graphs, then γ(G)γ 2/5|V|. We also characterize those connected graphs with γ(G)γ 2/5|V|.     

Even graphs

A nontrivial connected graph G is called even if for each vertex v of G there is a unique vertex v such that d(v, v ) = diam G. Special classes of even graphs are defined and compared to each other. In particular, an even graph G is called symmetric if d(u, v) + d(u, v ) = diam G for all u, v ∈ V(G). Several properties of even and symmetric even graphs are stated. For an even graph of order n and diameter d other than an even cycle it is shown that n ≥ 3d – 1 and conjectured that n ≥ 4d – 4. This conjecture is proved for symmetric even graphs and it is shown that for each pair of integers n, d with n even, d ≥ 2 and n ≥ 4d – 4 there exists an even graph of order n and diameter d. Several ways of constructing new even graphs from known ones are presented.