On p-intersection representations

For a graph G = (V,E) and integer p, a p-intersection representation is a family F = {S_x: × E V} of subsets of a set S with the property that |S_u ∩ S_ν| ≥ p ∩ {u, ν} E E. It is conjectured in [1] that θ_p(G) ≤ θ (K_n/2,n/2) (1 + o(1)) holds for any graph with n vertices. This is known to be true for p = 1 by [4]. In [1], θ (K_n/2,n/2) ≥ (n² + (2p− 1n)n)/4p is proved for any n and p. Here, we show that this is asymptotically best possible. Further, we provide a bound on θp(G) for all graphs with bounded degree. In particular, we prove θp(G) ≤ O(n^1/p) for any graph Gwith the maximum degree bounded by a constant. Finally, we also investigate the value of θ_p for trees. Improving on an earlier result of M. Jacobson, A. Kézdy, and D. West, (The 2-intersection number of paths and bounded-degree trees, preprint), we show that θ₂(T) ≤ O(d√n) for any tree T with maximum-degree d and θ₂(T) ≤ O(n^3/4) for any tree on n vertices. We conjecture that our results can be further improved and that θ₂(T) ≤ O(d√n) as long as Δ(T) ≤ √n. If this conjecture is true, our method gives θ₂(T) ≤ O(n^3/4) for any tree T which would be the best possible. © 1996 John Wiley & Sons, Inc.     

Claw‐decompositions and tutte‐orientations

We conjecture that, for each tree T, there exists a natural number k_T such that the following holds: If G is a k_T‐edge‐connected graph such that |E(T)| divides |E(G)|, then the edges of G can be divided into parts, each of which is isomorphic to T. We prove that for T = K_1,3 (the claw), this holds if and only if there exists a (smallest) natural number k_t such that every k_t‐edge‐connected graph has an orientation for which the indegree of each vertex equals its outdegree modulo 3. Tutte's 3‐flow conjecture says that k_t = 4. We prove the weaker statement that every 4$\lceil$ log n$\rceil$ ‐edge‐connected graph with n vertices has an edge‐decomposition into claws provided its number of edges is divisible by 3. We also prove that every triangulation of a surface has an edge‐decomposition into claws. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 135–146, 2006     

Lower bounds on signed edge total domination numbers in graphs

The open neighborhood N _G (e) of an edge e in a graph G is the set consisting of all edges having a common end-vertex with e. Let f be a function on E(G), the edge set of G, into the set {−1, 1}. If for each e ∈ E(G), then f is called a signed edge total dominating function of G. The minimum of the values , taken over all signed edge total dominating function f of G, is called the signed edge total domination number of G and is denoted by γ _st ′(G). Obviously, γ _st ′(G) is defined only for graphs G which have no connected components isomorphic to K ₂. In this paper we present some lower bounds for γ _st ′(G). In particular, we prove that γ _st ′(T) ⩾ 2 − m/3 for every tree T of size m ⩾ 2. We also classify all trees T with γ _st ′(T). Research supported by a Faculty Research Grant, University of West Georgia.     

Fan‐type theorem for path‐connectivity

For a connected noncomplete graph G, let μ(G):=min{max {d_G(u), d_G(v)}:d_G(u, v)=2}. A well‐known theorem of Fan says that every 2‐connected noncomplete graph has a cycle of length at least min{|V(G)|, 2μ(G)}. In this paper, we prove the following Fan‐type theorem: if G is a 3‐connected noncomplete graph, then each pair of distinct vertices of G is joined by a path of length at least min{|V(G)|?1, 2μ(G)?2}. As consequences, we have: (i) if G is a 3‐connected noncomplete graph with , then G is Hamilton‐connected; (ii) if G is a (s+2)‐connected noncomplete graph, where s≥1 is an integer, then through each path of length s of G there passes a cycle of length≥min{|V(G)|, 2μ(G)?s}. Several results known before are generalized and a conjecture of Enomoto, Hirohata, and Ota is proved. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 265–282, 2002 DOI 10.1002/jgt.10028     

Characterization of $ {\mathbb{A}_{16}} $ by a noncommuting graph

Let G be a finite non-Abelian group. We define a graph Γ _G ; called the noncommuting graph of G; with a vertex set G − Z(G) such that two vertices x and y are adjacent if and only if xy ≠ yx: Abdollahi, Akbari, and Maimani put forward the following conjecture (the AAM conjecture): If S is a finite non-Abelian simple group and G is a group such that Γ _S ≅ Γ _G ; then S ≅ G: It is still unknown if this conjecture holds for all simple finite groups with connected prime graph except \mathbbA₁₀ {\mathbb{A}_{10}} , L ₄(8), L ₄(4), and U ₄(4). In this paper, we prove that if \mathbbA₁₆ {\mathbb{A}_{16}} denotes the alternating group of degree 16; then, for any finite group G; the graph isomorphism G_\mathbbA16 @ G_G {\Gamma_{{\mathbb{A}_{16}}}} \cong {\Gamma_G} implies that \mathbbA₁₆ @ G {\mathbb{A}_{16}} \cong G .     

Thomassen's conjecture implies polynomiality of 1‐Hamilton‐connectedness in line graphs

A graph G is 1‐Hamilton‐connected if G?x is Hamilton‐connected for every x∈V(G), and G is 2‐edge‐Hamilton‐connected if the graph G+ X has a hamiltonian cycle containing all edges of X for any X?E⁺(G) = {xy| x, y∈V(G)} with 1≤|X|≤2. We prove that Thomassen's conjecture (every 4‐connected line graph is hamiltonian, or, equivalently, every snark has a dominating cycle) is equivalent to the statements that every 4‐connected line graph is 1‐Hamilton‐connected and/or 2‐edge‐Hamilton‐connected. As a corollary, we obtain that Thomassen's conjecture implies polynomiality of both 1‐Hamilton‐connectedness and 2‐edge‐Hamilton‐connectedness in line graphs. Consequently, proving that 1‐Hamilton‐connectedness is NP‐complete in line graphs would disprove Thomassen's conjecture, unless P = NP. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 241–250, 2012     

Combinatorial interpretations for TG(1, −1)

Let G be a connected and simple graph with vertex set {1, 2, …, n + 1} and T_G(x, y) the Tutte polynomial of G. In this paper, we give combinatorial interpretations for T_G(1, ?1). In particular, we give the definitions of even spanning tree and left spanning tree. We prove T_G(1, ?1) is the number of even‐left spanning trees of G. We associate a permutation with a spanning forest of G and give the definition of odd G‐permutations. We show T_G(1, ?1) is the number of odd G‐permutations. We give a bijection from the set of odd K_{n + 1}‐permutations to the set of alternating permutations on the set {1, 2, …, n}. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 341–348, 2012     

Uniquely Edge-3-Colorable Graphs and Snarks

 A cubic graph G is uniquely edge-3-colorable if G has precisely one 1-factorization. It is proved in this paper, if a uniquely edge-3-colorable, cubic graph G is cyclically 4-edge-connected, but not cyclically 5-edge-connected, then G must contain a snark as a minor. This is an approach to a conjecture that every triangle free uniquely edge-3-colorable cubic graph must have the Petersen graph as a minor. Fiorini and Wilson (1976) conjectured that every uniquely edge-3-colorable planar cubic graph must have a triangle. It is proved in this paper that every counterexample to this conjecture is cyclically 5-edge-connected and that in a minimal counterexample to the conjecture, every cyclic 5-edge-cut is trivial (an edge-cut T of G is cyclic if no component of G\T is acyclic and a cyclic edge-cut T is trivial if one component of G\T is a circuit of length |T|). Received: July 14, 1997 Revised: June 11, 1998     

The Edge Correlation of Random Forests

The conjecture was made by Kahn that a spanning forest F chosen uniformly at random from all forests of any finite graph G has the edge-negative association property. If true, the conjecture would mean that given any two edges ε₁ and ε₂ in G, the inequality \mathbbP(e₁ ? F, e₂ ? F) ￡ \mathbbP(e₁ ? F)\mathbbP(e₂ ? F){{\mathbb{P}(\varepsilon_{1} \in \mathbf{F}, \varepsilon_{2} \in \mathbf{F}) \leq \mathbb{P}(\varepsilon_{1} \in \mathbf{F})\mathbb{P}(\varepsilon_{2} \in \mathbf{F})}} would hold. We use enumerative methods to show that this conjecture is true for n large enough when G is a complete graph on n vertices. We derive explicit related results for random trees.     

An application of the combinatorial Nullstellensatz to a graph labelling problem

An antimagic labelling of a graph G with m edges and n vertices is a bijection from the set of edges of G to the set of integers {1,…,m}, such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with that vertex. A graph is called antimagic if it admits an antimagic labelling. In N. Hartsfield and G. Ringle, Pearls in Graph Theory, Academic Press, Inc., Boston, 1990, Ringel has conjectured that every simple connected graph, other than K₂, is antimagic. In this article, we prove a special case of this conjecture. Namely, we prove that if G is a graph on n=p^k vertices, where p is an odd prime and k is a positive integer that admits a C_p‐factor, then it is antimagic. The case p=3 was proved in D. Hefetz, J Graph Theory 50 (2005), 263–272. Our main tool is the combinatorial Nullstellensatz [N. Alon, Combin Probab Comput 8(1–2) (1999), 7–29]. © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 70–82, 2010.     

A reduction theorem for the McKay conjecture

The McKay conjecture asserts that for every finite group G and every prime p, the number of irreducible characters of G having p’-degree is equal to the number of such characters of the normalizer of a Sylow p-subgroup of G. Although this has been confirmed for large numbers of groups, including, for example, all solvable groups and all symmetric groups, no general proof has yet been found. In this paper, we reduce the McKay conjecture to a question about simple groups. We give a list of conditions that we hope all simple groups will satisfy, and we show that the McKay conjecture will hold for a finite group G if every simple group involved in G satisfies these conditions. Also, we establish that our conditions are satisfied for the simple groups PSL₂(q) for all prime powers q≥4, and for the Suzuki groups Sz(q) and Ree groups R(q), where q=2 ^e or q=3 ^e respectively, and e>1 is odd. Since our conditions are also satisfied by the sporadic simple group J ₁, it follows that the McKay conjecture holds (for all primes p) for every finite group having an abelian Sylow 2-subgroup.     

On antimagic directed graphs

An antimagic labeling of an undirected graph G with n vertices and m edges is a bijection from the set of edges of G to the integers {1, …, m} such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with that vertex. A graph is called antimagic if it admits an antimagic labeling. In (N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, Boston, 1990, pp. 108–109), Hartsfield and Ringel conjectured that every simple connected graph, other than K₂, is antimagic. Despite considerable effort in recent years, this conjecture is still open. In this article we study a natural variation; namely, we consider antimagic labelings of directed graphs. In particular, we prove that every directed graph whose underlying undirected graph is “dense” is antimagic, and that almost every undirected d‐regular graph admits an orientation which is antimagic. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 219–232, 2010     

The Erdős-Sós Conjecture for Graphs Whose Complements Contain No <Emphasis Type="Italic">C</Emphasis><Subscript>4</Subscript>

Abstract Erdős and Sós conjectured in 1963 (see [1], Problem 12 in 247) that every graph G on n vertices with size contains every tree T of size k. In this paper, we prove the conjecture for graphs whose complements contain no cycles of length 4. Supported by the National Natural Science Foundation of China (No.19971086) and the Doctoral Foundation of Hainan University.     

Dirac's Condition for Completely Independent Spanning Trees

Two spanning trees T₁ and T₂ of a graph G are completely independent if, for any two vertices u and v, the paths from u to v in T₁ and T₂ are internally disjoint. In this article, we show two sufficient conditions for the existence of completely independent spanning trees. First, we show that a graph of n vertices has two completely independent spanning trees if the minimum degree of the graph is at least . Then, we prove that the square of a 2‐connected graph has two completely independent spanning trees. These conditions are known to be sufficient conditions for Hamiltonian graphs.     

Odd‐angulated graphs and cancelling factors in box products

Under what conditions is it true that if there is a graph homomorphism G □ H → G □ T, then there is a graph homomorphism H→ T? Let G be a connected graph of odd girth 2k + 1. We say that G is (2k + 1)‐angulated if every two vertices of G are joined by a path each of whose edges lies on some (2k + 1)‐cycle. We call G strongly (2k + 1)‐angulated if every two vertices are connected by a sequence of (2k + 1)‐cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2k + 1)‐angulated, H is any graph, S, T are graphs with odd girth at least 2k + 1, and ?: G□ H→S□T is a graph homomorphism, then either ? maps G□{h} to S□{t_h} for all h∈V(H) where t_h∈V(T) depends on h; or ? maps G□{h} to {s_h}□ T for all h∈V(H) where s_h∈V(S) depends on h. This theorem allows us to prove several sufficient conditions for a cancelation law of a graph homomorphism between two box products with a common factor. We conclude the article with some open questions. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:221‐238, 2008     

Induced trees in graphs of large chromatic number

Gyárfás and Sumner independently conjectured that for every tree T and integer k there is an integer f(k, T) such that every graph G with χ(G) > f(k, t) contains either K_k or an induced copy of T. We prove a ‘topological’ version of the conjecture: for every tree T and integer k there is g(k,T) such that every graph G with χ(G) > g(k,t) contains either K_k or an induced copy of a subdivision of T. © 1997 John Wiley & Sons, Inc.     

Stability for large forbidden subgraphs

In this note we strengthen the stability theorem of Erd?s and Simonovits. Write K_r(s₁, …, s_r) for the complete r‐partite graph with classes of sizes s₁, …, s_r and T_r(n) for the r‐partite Turán graph of order n. Our main result is: For all r≥2 and all sufficiently small c>0, ε>0, every graph G of sufficiently large order n with e(G)>(1?1/r?ε)n²/2 satisfies one of the conditions:

• (a) G contains a $K_{r+1} (\lfloor c\,\mbox{ln}\,n \rfloor,\ldots,\lfloor c\,\mbox{ln}\,n \rfloor,\lceil n^{{1}-\sqrt{c}}\rceil)In this note we strengthen the stability theorem of Erd?s and Simonovits. Write K_r(s₁, …, s_r) for the complete r‐partite graph with classes of sizes s₁, …, s_r and T_r(n) for the r‐partite Turán graph of order n. Our main result is: For all r≥2 and all sufficiently small c>0, ε>0, every graph G of sufficiently large order n with e(G)>(1?1/r?ε)n²/2 satisfies one of the conditions:
  • (a) G contains a $K_{r+1} (\lfloor c\,\mbox{ln}\,n \rfloor,\ldots,\lfloor c\,\mbox{ln}\,n \rfloor,\lceil n^{{1}-\sqrt{c}}\rceil)$;
  • (b) G differs from T_r(n) in fewer than (ε^1/3+c^1/(3r+3))n² edges.
  Letting µ(G) be the spectral radius of G, we prove also a spectral stability theorem: For all r≥2 and all sufficiently small c>0, ε>0, every graph G of sufficiently large order n with µ(G)>(1?1/r?ε)n satisfies one of the conditions:
  • (a) G contains a $K_{r+1}(\lfloor c\,\mbox{ln}\,n\rfloor,\ldots,\lfloor c\,\mbox{ln}\,n\rfloor,\lceil n^{1-\sqrt{c}}\rceil)$;
  • (b) G differs from T_r(n) in fewer than (ε^1/4+c^1/(8r+8))n² edges.
  © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 362–368, 2009     

Balanced judicious bipartitions of graphs

A bipartition of the vertex set of a graph is called balanced if the sizes of the sets in the bipartition differ by at most one. B. Bollobás and A. D. Scott, Random Struct Alg 21 (2002), 414–430 conjectured that if G is a graph with minimum degree of at least 2 then V(G) admits a balanced bipartition V₁, V₂ such that for each i, G has at most |E(G)|/3 edges with both ends in V_i. The minimum degree condition is necessary, and a result of B. Bollobás and A. D. Scott, J. Graph Theory 46 (2004), 131–143 shows that this conjecture holds for regular graphs G(i.e., when Δ(G)=δ(G)). We prove this conjecture for graphs G with \begin{eqnarray*}\Delta(G)\le\frac{7}{5}\delta(G)\end{eqnarray*}; hence, it holds for graphs ]ensuremathG with \begin{eqnarray*}\delta(G)\ge\frac{5}{7}|V(G)|\end{eqnarray*}. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 210–225, 2010     

On covering vertices of a graph by trees

The purpose of this paper is to initiate study of the following problem: Let G be a graph, and k?1. Determine the minimum number s of trees T₁,…,T_s, Δ(T_i)?k,i=1,…,s, covering all vertices of G. We conjecture: Let G be a connected graph, and k?2. Then the vertices of G can be covered by edge-disjoint trees of maximum degree ?k. As a support for the conjecture we prove the statement for some values of δ and k.     

The 2-intersection number of paths and bounded-degree trees

We represent a graph by assigning each vertex a finite set such that vertices are adjacent if and only if the corresponding sets have at least two common elements. The 2-intersection number θ₂(G) of a graph G is the minimum size of the union of sets in such a representation. We prove that the maximum order of a path that can be represented in this way using t elements is between (t² - 19t + 4)/4 and (t² - t + 6)/4, making θ₂(P_n) asymptotic to 2√n. We also show the existence of a constant c depending on ? such that, for any tree T with maximum degree at most d, θ₂(T) ≤ c(√n)^1+?. When the maximum degree is not bounded, there is an n-vertex tree T with θ₂(T) > .945n^2/3. © 1995 John Wiley & Sons, Inc.