Note on Perfect Forests

A spanning subgraph F of a graph G is called perfect if F is a forest, the degree of each vertex x in F is odd, and each tree of F is an induced subgraph of G. We provide a short linear‐algebraic proof of the following theorem of A. D. Scott (Graphs Combin 17 (2001), 539–553): A connected graph G contains a perfect forest if and only if G has an even number of vertices.     

On Perfect Matching Coverings and Even Subgraph Coverings

A perfect matching covering of a graph G is a set of perfect matchings of G such that every edge of G is contained in at least one member of it. Berge conjectured that every bridgeless cubic graph admits a perfect matching covering of order at most 5 (we call such a collection of perfect matchings a Berge covering of G). A cubic graph G is called a Kotzig graph if G has a 3‐edge‐coloring such that each pair of colors forms a hamiltonian circuit (introduced by R. Häggkvist, K. Markström, J Combin Theory Ser B 96 (2006), 183–206). In this article, we prove that if there is a vertex w of a cubic graph G such that , the graph obtained from by suppressing all degree two vertices is a Kotzig graph, then G has a Berge covering. We also obtain some results concerning the so‐called 5‐even subgraph double cover conjecture.     

Perfect Matchings of Regular Bipartite Graphs

Let G be a regular bipartite graph and . We show that there exist perfect matchings of G containing both, an odd and an even number of edges from X if and only if the signed graph , that is a graph G with exactly the edges from X being negative, is not equivalent to . In fact, we prove that for a given signed regular bipartite graph with minimum signature, it is possible to find perfect matchings that contain exactly no negative edges or an arbitrary one preselected negative edge. Moreover, if the underlying graph is cubic, there exists a perfect matching with exactly two preselected negative edges. As an application of our results we show that each signed regular bipartite graph that contains an unbalanced circuit has a 2‐cycle‐cover such that each cycle contains an odd number of negative edges.     

Flows and Bisections in Cubic Graphs

A k‐weak bisection of a cubic graph G is a partition of the vertex‐set of G into two parts V₁ and V₂ of equal size, such that each connected component of the subgraph of G induced by () is a tree of at most vertices. This notion can be viewed as a relaxed version of nowhere‐zero flows, as it directly follows from old results of Jaeger that every cubic graph G with a circular nowhere‐zero r‐flow has a ‐weak bisection. In this article, we study problems related to the existence of k‐weak bisections. We believe that every cubic graph that has a perfect matching, other than the Petersen graph, admits a 4‐weak bisection and we present a family of cubic graphs with no perfect matching that do not admit such a bisection. The main result of this article is that every cubic graph admits a 5‐weak bisection. When restricted to bridgeless graphs, that result would be a consequence of the assertion of the 5‐flow Conjecture and as such it can be considered a (very small) step toward proving that assertion. However, the harder part of our proof focuses on graphs that do contain bridges.     

Uniquely Cycle‐Saturated Graphs

Given a graph F, a graph G is uniquely F‐saturated if F is not a subgraph of G and adding any edge of the complement to G completes exactly one copy of F. In this article, we study uniquely ‐saturated graphs. We prove the following: (1) a graph is uniquely C₅‐saturated if and only if it is a friendship graph. (2) There are no uniquely C₆‐saturated graphs or uniquely C₇‐saturated graphs. (3) For , there are only finitely many uniquely ‐saturated graphs (we conjecture that in fact there are none). Additionally, our results show that there are finitely many k‐friendship graphs (as defined by Kotzig) for .     

Triply Existentially Complete Triangle‐Free Graphs

A triangle‐free graph G is called k‐existentially complete if for every induced k‐vertex subgraph H of G, every extension of H to a ‐vertex triangle‐free graph can be realized by adding another vertex of G to H. Cherlin  11 , 12 asked whether k‐existentially complete triangle‐free graphs exist for every k. Here, we present known and new constructions of 3‐existentially complete triangle‐free graphs.     

On the Probability that a Random Subgraph Contains a Circuit

Let and . We show that, if G is a sufficiently large simple graph of average degree at least μ, and H is a random spanning subgraph of G formed by including each edge independently with probability , then H contains a cycle with probability at least .     

On the Iterated Biclique Operator

A biclique of a graph G is a maximal induced complete bipartite subgraph of G. The biclique graph of G, denoted by , is the intersection graph of the bicliques of G. We say that a graph G diverges (or converges or is periodic) under an operator F whenever ( for some m, or for some k and , respectively). Given a graph G, the iterated biclique graph of G, denoted by , is the graph obtained by applying the biclique operator k successive times to G. In this article, we study the iterated biclique graph of G. In particular, we classify the different behaviors of when the number of iterations k grows to infinity. That is, we prove that a graph either diverges or converges under the biclique operator. We give a forbidden structure characterization of convergent graphs, which yield a polynomial time algorithm to decide if a given graph diverges or converges. This is in sharp contrast with the situsation for the better known clique operator, where it is not even known if the corresponding problem is decidable. © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 181–190, 2013     

A Note on Degenerate and Spectrally Degenerate Graphs

A graph G is called spectrally d‐degenerate if the largest eigenvalue of each subgraph of it with maximum degree D is at most . We prove that for every constant M there is a graph with minimum degree M, which is spectrally 50‐degenerate. This settles a problem of Dvo?ák and Mohar (Spectrally degenerate graphs: Hereditary case, arXiv: 1010.3367).     

Independent Domination in Cubic Graphs

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number of G, denoted by , is the minimum cardinality of an independent dominating set. In this article, we show that if is a connected cubic graph of order n that does not have a subgraph isomorphic to K_{2, 3}, then . As a consequence of our main result, we deduce Reed's important result [Combin Probab Comput 5 (1996), 277–295] that if G is a cubic graph of order n, then , where denotes the domination number of G.     

Outerplanar and Planar Oriented Cliques

The clique number of an undirected graph G is the maximum order of a complete subgraph of G and is a well‐known lower bound for the chromatic number of G. Every proper k‐coloring of G may be viewed as a homomorphism (an edge‐preserving vertex mapping) of G to the complete graph of order k. By considering homomorphisms of oriented graphs (digraphs without cycles of length at most 2), we get a natural notion of (oriented) colorings and oriented chromatic number of oriented graphs. An oriented clique is then an oriented graph whose number of vertices and oriented chromatic number coincide. However, the structure of oriented cliques is much less understood than in the undirected case. In this article, we study the structure of outerplanar and planar oriented cliques. We first provide a list of 11 graphs and prove that an outerplanar graph can be oriented as an oriented clique if and only if it contains one of these graphs as a spanning subgraph. Klostermeyer and MacGillivray conjectured that the order of a planar oriented clique is at most 15, which was later proved by Sen. We show that any planar oriented clique on 15 vertices must contain a particular oriented graph as a spanning subgraph, thus reproving the above conjecture. We also provide tight upper bounds for the order of planar oriented cliques of girth k for all .     

Minimal Obstructions for 1‐Immersions and Hardness of 1‐Planarity Testing

A graph is 1‐planar if it can be drawn on the plane so that each edge is crossed by no more than one other edge (and any pair of crossing edges cross only once). A non‐1‐planar graph G is minimal if the graph is 1‐planar for every edge e of G. We construct two infinite families of minimal non‐1‐planar graphs and show that for every integer , there are at least nonisomorphic minimal non‐1‐planar graphs of order n. It is also proved that testing 1‐planarity is NP‐complete.     

Perfect Graphs with No Balanced Skew‐Partition are 2‐Clique‐Colorable

A graph G is perfect if for all induced subgraphs H of G, . A graph G is Berge if neither G nor its complement contains an induced odd cycle of length at least five. The Strong Perfect Graph Theorem [9] states that a graph is perfect if and only if it is Berge. The Strong Perfect Graph Theorem was obtained as a consequence of a decomposition theorem for Berge graphs [M. Chudnovsky, Berge trigraphs and their applications, PhD thesis, Princeton University, 2003; M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Ann Math 164 (2006), 51–229.], and one of the decompositions in this decomposition theorem was the “balanced skew‐partition.” A clique‐coloring of a graph G is an assignment of colors to the vertices of G in such a way that no inclusion‐wise maximal clique of G of size at least two is monochromatic, and the clique‐chromatic number of G, denoted by , is the smallest number of colors needed to clique‐color G. There exist graphs of arbitrarily large clique‐chromatic number, but it is not known whether the clique‐chromatic number of perfect graphs is bounded. In this article, we prove that every perfect graph that does not admit a balanced skew‐partition is 2‐clique colorable. The main tool used in the proof is a decomposition theorem for “tame Berge trigraphs” due to Chudnovsky et al. ( http://arxiv.org/abs/1308.6444 ).     

χ‐bounded families of oriented graphs

A famous conjecture of Gyárfás and Sumner states for any tree T and integer k, if the chromatic number of a graph is large enough, either the graph contains a clique of size k or it contains T as an induced subgraph. We discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for every oriented star S and integer k, if the chromatic number of a digraph is large enough, either the digraph contains a clique of size k or it contains S as an induced subgraph. As an evidence, we prove that for any oriented star S, every oriented graph with sufficiently large chromatic number contains either a transitive tournament of order 3 or S as an induced subdigraph. We then study for which sets of orientations of P₄ (the path on four vertices) similar statements hold. We establish some positive and negative results.     

Constructing a Family of 4‐Critical Planar Graphs with High Edge Density

A graph is a k‐critical graph if G is not ‐colorable but every proper subgraph of G is ‐colorable. In this article, we construct a family of 4‐critical planar graphs with n vertices and edges. As a consequence, this improves the bound for the maximum edge density attained by Abbott and Zhou. We conjecture that this is the largest edge density for a 4‐critical planar graph.     

Vizing's 2‐Factor Conjecture Involving Large Maximum Degree

Let G be an n‐vertex simple graph, and let and denote the maximum degree and chromatic index of G, respectively. Vizing proved that or . Define G to be Δ‐critical if and for every proper subgraph H of G. In 1965, Vizing conjectured that if G is an n‐vertex Δ‐critical graph, then G has a 2‐factor. Luo and Zhao showed if G is an n‐vertex Δ‐critical graph with , then G has a hamiltonian cycle, and so G has a 2‐factor. In this article, we show that if G is an n‐vertex Δ‐critical graph with , then G has a 2‐factor.     

Pancyclic subgraphs of random graphs

An n‐vertex graph is called pancyclic if it contains a cycle of length t for all 3≤t≤n. In this article, we study pancyclicity of random graphs in the context of resilience, and prove that if p>n^?1/2, then the random graph G(n, p) a.a.s. satisfies the following property: Every Hamiltonian subgraph of G(n, p) with more than edges is pancyclic. This result is best possible in two ways. First, the range of p is asymptotically tight; second, the proportion of edges cannot be reduced. Our theorem extends a classical theorem of Bondy, and is closely related to a recent work of Krivelevich et al. The proof uses a recent result of Schacht (also independently obtained by Conlon and Gowers). © 2011 Wiley Periodicals, Inc.     

Equimatchable Graphs on Surfaces

A graph G is equimatchable if each matching in G is a subset of a maximum‐size matching and it is factor critical if has a perfect matching for each vertex v of G. It is known that any 2‐connected equimatchable graph is either bipartite or factor critical. We prove that for 2‐connected factor‐critical equimatchable graph G the graph is either or for some n for any vertex v of G and any minimal matching M such that is a component of . We use this result to improve the upper bounds on the maximum number of vertices of 2‐connected equimatchable factor‐critical graphs embeddable in the orientable surface of genus g to if and to if . Moreover, for any nonnegative integer g we construct a 2‐connected equimatchable factor‐critical graph with genus g and more than vertices, which establishes that the maximum size of such graphs is . Similar bounds are obtained also for nonorientable surfaces. In the bipartite case for any nonnegative integers g, h, and k we provide a construction of arbitrarily large 2‐connected equimatchable bipartite graphs with orientable genus g, respectively nonorientable genus h, and a genus embedding with face‐width k. Finally, we prove that any d‐degenerate 2‐connected equimatchable factor‐critical graph has at most vertices, where a graph is d‐degenerate if every its induced subgraph contains a vertex of degree at most d.     

The Erdös–Hajnal Conjecture—A Survey

The Erdös–Hajnal conjecture states that for every graph H, there exists a constant such that every graph G with no induced subgraph isomorphic to H has either a clique or a stable set of size at least . This article is a survey of some of the known results on this conjecture.     

A note on shortest cycle covers of cubic graphs

Let SCC₃(G) be the length of a shortest 3‐cycle cover of a bridgeless cubic graph G. It is proved in this note that if G contains no circuit of length 5 (an improvement of Jackson's (JCTB 1994) result: if G has girth at least 7) and if all 5‐circuits of G are disjoint (a new upper bound of SCC₃(G) for the special class of graphs).