Triangle-free graphs with no six-vertex induced path

The graphs with no five-vertex induced path are still not understood. But in the triangle-free case, we can do this and one better; we give an explicit construction for all triangle-free graphs with no six-vertex induced path. Here are three examples: the 16-vertex Clebsch graph, the graph obtained from an 8-cycle by making opposite vertices adjacent, and the graph obtained from a complete bipartite graph by subdividing a perfect matching. We show that every connected triangle-free graph with no six-vertex induced path is an induced subgraph of one of these three (modulo some twinning and duplication).     

A Hopf algebra for counting cycles

Cycles, also known as self-avoiding polygons, elementary circuits or simple cycles, are closed walks which are not allowed to visit any vertex more than once. We present an exact formula for enumerating such cycles of any length on any directed graph involving a sum over its induced subgraphs. This result stems from a Hopf algebra, which we construct explicitly, and which provides further means of counting cycles. Finally, we obtain a more general theorem asserting that any Lie idempotent can be used to enumerate cycles.     

Polyhedral study of the connected subgraph problem

In this paper, we study the connected subgraph polytope which is the convex hull of the solutions to a related combinatorial optimization problem called the maximum weight connected subgraph problem. We strengthen a cut-based formulation by considering some new partition inequalities for which we give necessary and sufficient conditions to be facet defining. Based on the separation problem associated with these inequalities, we give a complete polyhedral characterization of the connected subgraph polytope on cycles and trees.     

On a Conjecture of Erdős,Gallai, and Tuza

Erd?s, Gallai, and Tuza posed the following problem: given an n‐vertex graph G, let denote the smallest size of a set of edges whose deletion makes G triangle‐free, and let denote the largest size of a set of edges containing at most one edge from each triangle of G. Is it always the case that ? We have two main results. We first obtain the upper bound , as a partial result toward the Erd?s–Gallai–Tuza conjecture. We also show that always , where m is the number of edges in G; this bound is sharp in several notable cases.     

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     

Anti‐Ramsey numbers of doubly edge‐critical graphs

Given a graph H and a positive integer n, Anti‐Ramsey number AR(n, H) is the maximum number of colors in an edge‐coloring of K_n that contains no polychromatic copy of H. The anti‐Ramsey numbers were introduced in the 1970s by Erd?s, Simonovits, and Sós, who among other things, determined this function for cliques. In general, few exact values of AR(n, H) are known. Let us call a graph H doubly edge‐critical if χ(H?e)≥p+ 1 for each edge e∈E(H) and there exist two edges e₁, e₂ of H for which χ(H?e₁?e₂)=p. Here, we obtain the exact value of AR(n, H) for any doubly edge‐critical H when n?n₀(H) is sufficiently large. A main ingredient of our proof is the stability theorem of Erd?s and Simonovits for the Turán problem. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 210–218, 2009     

矩阵特征多项式的图论计算公式

给出了赋权有向图邻接矩阵特征多项式的图论计算公式,从而得到了一般矩阵特征多项式的图论计算方法,并且研究了赋权有向图邻接矩阵特征多项式和谱半径的一些性质．     

Factors and vertex-deleted subgraphs

A relationship is considered between an f-factor of a graph and that of its vertex-deleted subgraphs. Katerinis [Some results on the existence of 2n-factors in terms of vertex-deleted subgraphs, Ars Combin. 16 (1983) 271-277] proved that for even integer k, if G-x has a k-factor for each x∈V(G), then G has a k-factor. Enomoto and Tokuda [Complete-factors and f-factors, Discrete Math. 220 (2000) 239-242] generalized Katerinis’ result to f-factors, and proved that if G-x has an f-factor for each x∈V(G), then G has an f-factor for an integer-valued function f defined on V(G) with even. In this paper, we consider a similar problem to that of Enomoto and Tokuda, where for several vertices x we do not have to know whether G-x has an f-factor. Let G be a graph, X be a set of vertices, and let f be an integer-valued function defined on V(G) with even, |V(G)-X|?2. We prove that if and if G-x has an f-factor for each x∈V(G)-X, then G has an f-factor. Moreover, if G excludes an isolated vertex, then we can replace the condition with . Furthermore the condition will be when |X|=1.     

Turán’s theorem inverted

In this note we complete an investigation started by Erd?s in 1963 that aims to find the strongest possible conclusion from the hypothesis of Turán’s theorem in extremal graph theory.Let be the complete r-partite graph with parts of sizes s₁≥2,s₂,…,s_r with an edge added to the first part. Letting t_r(n) be the number of edges of the r-partite Turán graph of order n, we prove that:For all r≥2 and all sufficiently small c>0, every graph of sufficiently large order n with t_r(n)+1 edges contains a .We also give a corresponding stability theorem and two supporting results of wider scope.