A note on minimal matching covered graphs

A graph is called matching covered if for its every edge there is a maximum matching containing it. It is shown that minimal matching covered graphs without isolated vertices contain a perfect matching.     

On the well‐posedness of a mathematical model of quorum‐sensing in patchy biofilm communities

We analyze a system of reaction–diffusion equations that models quorum‐sensing in a growing biofilm. The model comprises two nonlinear diffusion effects: a porous medium‐type degeneracy and super diffusion. We prove the well‐posedness of the model. In particular, we present for the first time a uniqueness result for this type of problem. Moreover, we illustrate the behavior of model solutions in numerical simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

Tutte sets in graphs I: Maximal tutte sets and D‐graphs

A well‐known formula of Tutte and Berge expresses the size of a maximum matching in a graph G in terms of what is usually called the deficiency of G. A subset X of V(G) for which this deficiency is attained is called a Tutte set of G. While much is known about maximum matchings, less is known about the structure of Tutte sets. In this article, we study the structural aspects of maximal Tutte sets in a graph G. Towards this end, we introduce a related graph D(G). We first show that the maximal Tutte sets in G are precisely the maximal independent sets in its D‐graph D(G), and then continue with the study of D‐graphs in their own right, and of iterated D‐graphs. We show that G is isomorphic to a spanning subgraph of D(G), and characterize the graphs for which G?D(G) and for which D(G)?D²(G). Surprisingly, it turns out that for every graph G with a perfect matching, D³(G)?D²(G). Finally, we characterize bipartite D‐graphs and comment on the problem of characterizing D‐graphs in general. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 343–358, 2007     

Injective choosability of subcubic planar graphs with girth 6

An injective coloring of a graph $G$ is an assignment of colors to the vertices of $G$ so that any two vertices with a common neighbor have distinct colors. A graph $G$ is injectively $k$-choosable if for any list assignment $L$, where $\left|L\left(v\right)\right|\ge k$ for all $v\in V\left(G\right)$, $G$ has an injective $L$-coloring. Injective colorings have applications in the theory of error-correcting codes and are closely related to other notions of colorability. In this paper, we show that subcubic planar graphs with girth at least 6 are injectively 5-choosable. This strengthens the result of Lu?ar, ?krekovski, and Tancer that subcubic planar graphs with girth at least 7 are injectively 5-colorable. Our result also improves several other results in particular cases.     

A family of regular graphs of girth 5

Murty [A generalization of the Hoffman-Singleton graph, Ars Combin. 7 (1979) 191-193.] constructed a family of (p^m+2)-regular graphs of girth five and order 2p^2m, where p?5 is a prime, which includes the Hoffman-Singleton graph [A.J. Hoffman, R.R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. (1960) 497-504]. This construction gives an upper bound for the least number f(k) of vertices of a k-regular graph with girth 5. In this paper, we extend the Murty construction to k-regular graphs with girth 5, for each k. In particular, we obtain new upper bounds for f(k), k?16.     

A characterization of P4‐indifference graphs

A graph is a P₄‐indifference graph if it admits a linear ordering ≺ on its vertices such that every chordless path with vertices a, b, c, d and edges ab, bc, cd has either a ≺ b ≺ c ≺ d or d ≺ c ≺ b ≺ a. P₄‐indifference graphs generalize indifference graphs and are perfectly orderable. We give a characterization of P₄‐indifference graphs by forbidden induced subgraphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 155‐162, 1999     

On k‐ordered graphs

Ng and Schultz [J Graph Theory 1 ( 6 ), 45–57] introduced the idea of cycle orderability. For a positive integer k, a graph G is k‐ordered if for every ordered sequence of k vertices, there is a cycle that encounters the vertices of the sequence in the given order. If the cycle is also a Hamiltonian cycle, then G is said to be k‐ordered Hamiltonian. We give sum of degree conditions for nonadjacent vertices and neighborhood union conditions that imply a graph is k‐ordered Hamiltonian. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 69–82, 2000     

On integro‐differential inclusions with operator‐valued kernels

We study integro‐differential inclusions in Hilbert spaces with operator‐valued kernels and give sufficient conditions for the well‐posedness. We show that several types of integro‐differential equations and inclusions are covered by the class of evolutionary inclusions, and we therefore give criteria for the well‐posedness within this framework. As an example, we apply our results to the equations of visco‐elasticity and to a class of nonlinear integro‐differential inclusions describing phase transition phenomena in materials with memory. Copyright © 2014 John Wiley & Sons, Ltd.     

Clique‐inverse graphs of K3‐free and K4‐free graphs

The clique graph K(G) of a given graph G is the intersection graph of the collection of maximal cliques of G. Given a family ℱ of graphs, the clique‐inverse graphs of ℱ are the graphs whose clique graphs belong to ℱ. In this work, we describe characterizations for clique‐inverse graphs of K₃‐free and K₄‐free graphs. The characterizations are formulated in terms of forbidden induced subgraphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 257–272, 2000     

T‐joins intersecting small edge‐cuts in graphs

In an earlier paper 3 , we studied cycles in graphs that intersect all edge‐cuts of prescribed sizes. Passing to a more general setting, we examine the existence of T‐joins in grafts that intersect all edge‐cuts whose size is in a given set A ?{1,2,3}. In particular, we characterize all the contraction‐minimal grafts admitting no T‐joins that intersect all edge‐cuts of size 1 and 2. We also show that every 3‐edge‐connected graft admits a T‐join intersecting all 3‐edge‐cuts. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 64–71, 2007     

Tree‐width and circumference of graphs

We prove that every graph of circumference k has tree‐width at most k ? 1 and that this bound is best possible. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 24–25, 2003     

Constructing cubic edge‐ but not vertex‐transitive graphs

An infinite family of cubic edge‐transitive but not vertex‐transitive graphs with edge stabilizer isomorphic to ℤ₂ is constructed. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 152–160, 2000     

One‐factorizations of complete graphs with vertex‐regular automorphism groups

We consider one‐factorizations of K_2n possessing an automorphism group acting regularly (sharply transitively) on vertices. We present some upper bounds on the number of one‐factors which are fixed by the group; further information is obtained when equality holds in these bounds. The case where the group is dihedral is studied in some detail, with some non‐existence statements in case the number of fixed one‐factors is as large as possible. Constructions both for dihedral groups and for some classes of abelian groups are given. © 2002 John Wiley & Sons, Inc. J Combin Designs 10: 1–16, 2002     

The Erd s-Sós conjecture for graphs of girth 5

We prove that every graph of girth at least 5 with minimum degree δ k/2 contains every tree with k edges, whose maximum degree does not exceed the maximum degree of the graph. An immediate consequence is that the famous Erd s-Sós Conjecture, saying that every graph of order n with more than n(k − 1)/2 edges contains every tree with k edges, is true for graphs of girth at least 5.     

Small vertex‐transitive and Cayley graphs of girth six and given degree: an algebraic approach

We examine the existing constructions of the smallest known vertex‐transitive graphs of a given degree and girth 6. It turns out that most of these graphs can be described in terms of regular lifts of suitable quotient graphs. A further outcome of our analysis is a precise identification of which of these graphs are Cayley. We also investigate higher level of transitivity of the smallest known vertex‐transitive graphs of a given degree and girth 6 and relate their constructions to near‐difference sets. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:265‐284, 2011     

A proof of a conjecture on the Randić index of graphs with given girth

The Randić index R(G) of a graph G is defined by , where is the degree of a vertex u in G and the summation extends over all edges uv of G. Aouchiche, Hansen and Zheng proposed the following conjecture: For any connected graph on n≥3 vertices with Randić index R and girth g,

with equalities if and only if . This paper is devoted to giving a confirmative proof to this conjecture.     

Precoloring extension for K4‐minor‐free graphs

Let G=(V, E) be a graph where every vertex v∈V is assigned a list of available colors L(v). We say that G is list colorable for a given list assignment if we can color every vertex using its list such that adjacent vertices get different colors. If L(v)={1, …, k} for all v∈V then a corresponding list coloring is nothing other than an ordinary k‐coloring of G. Assume that W?V is a subset of V such that G[W] is bipartite and each component of G[W] is precolored with two colors taken from a set of four. The minimum distance between the components of G[W] is denoted by d(W). We will show that if G is K₄‐minor‐free and d(W)≥7, then such a precoloring of W can be extended to a 4‐coloring of all of V. This result clarifies a question posed in 10. Moreover, we will show that such a precoloring is extendable to a list coloring of G for outerplanar graphs, provided that |L(v)|=4 for all v∈V\W and d(W)≥7. In both cases the bound for d(W) is best possible. © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 284‐294, 2009     

A conjecture on the algebraic connectivity of connected graphs with fixed girth

In this paper, we completely solve a conjecture on the minimum algebraic connectivity of connected graphs with fixed girth (see [S. Fallat, S. Kirkland, Extremizing algebraic connectivity subject to graph theoretic constraints, Electron. J. Linear Algebra 3 (1998) 48-74]).     

On factors of 4‐connected claw‐free graphs*

We consider the existence of several different kinds of factors in 4‐connected claw‐free graphs. This is motivated by the following two conjectures which are in fact equivalent by a recent result of the third author. Conjecture 1 (Thomassen): Every 4‐connected line graph is hamiltonian, i.e., has a connected 2‐factor. Conjecture 2 (Matthews and Sumner): Every 4‐connected claw‐free graph is hamiltonian. We first show that Conjecture 2 is true within the class of hourglass‐free graphs, i.e., graphs that do not contain an induced subgraph isomorphic to two triangles meeting in exactly one vertex. Next we show that a weaker form of Conjecture 2 is true, in which the conclusion is replaced by the conclusion that there exists a connected spanning subgraph in which each vertex has degree two or four. Finally we show that Conjectures 1 and 2 are equivalent to seemingly weaker conjectures in which the conclusion is replaced by the conclusion that there exists a spanning subgraph consisting of a bounded number of paths © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 125–136, 2001