Note on Perfect Forests in Digraphs

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. Alex Scott (Graphs Combin 17 (2001), 539–553) proved that every connected graph G contains a perfect forest if and only if G has an even number of vertices. We consider four generalizations to directed graphs of the concept of a perfect forest. While the problem of existence of the most straightforward one is NP‐hard, for the three others this problem is polynomial‐time solvable. Moreover, every digraph with only one strong component contains a directed forest of each of these three generalization types. One of our results extends Scott's theorem to digraphs in a nontrivial way.     

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 .     

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     

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.     

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.     

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 .     

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.     

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.     

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.     

The Maximum Number of Dominating Induced Matchings

A matching M of a graph G is a dominating induced matching (DIM) of G if every edge of G is either in M or adjacent with exactly one edge in M. We prove sharp upper bounds on the number of DIMs of a graph G and characterize all extremal graphs. Our results imply that if G is a graph of order n, then ; provided G is triangle‐free; and provided and G is connected.     

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 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.     

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 .     

A greedy algorithm for finding a large 2‐matching on a random cubic graph

A 2‐matching of a graph G is a spanning subgraph with maximum degree two. The size of a 2‐matching U is the number of edges in U and this is at least where n is the number of vertices of G and κ denotes the number of components. In this article, we analyze the performance of a greedy algorithm 2greedy for finding a large 2‐matching on a random 3‐regular graph. We prove that with high probability, the algorithm outputs a 2‐matching U with .     

Exponentially many 4‐list‐colorings of triangle‐free graphs on surfaces

Thomassen proved that every planar graph G on n vertices has at least distinct L‐colorings if L is a 5‐list‐assignment for G and at least distinct L‐colorings if L is a 3‐list‐assignment for G and G has girth at least five. Postle and Thomas proved that if G is a graph on n vertices embedded on a surface Σ of genus g, then there exist constants such that if G has an L‐coloring, then G has at least distinct L‐colorings if L is a 5‐list‐assignment for G or if L is a 3‐list‐assignment for G and G has girth at least five. More generally, they proved that there exist constants such that if G is a graph on n vertices embedded in a surface Σ of fixed genus g, H is a proper subgraph of G, and ϕ is an L‐coloring of H that extends to an L‐coloring of G, then ϕ extends to at least distinct L‐colorings of G if L is a 5‐list‐assignment or if L is a 3‐list‐assignment and G has girth at least five. We prove the same result if G is triangle‐free and L is a 4‐list‐assignment of G, where , and .     

A constructive characterisation of circuits in the simple (2,1)‐sparse matroid

A simple graph is a (2, 1)‐circuit if and for every proper subgraph H of G. Motivated, in part, by ongoing work to understand unique realisations of graphs on surfaces, we derive a constructive characterisation of (2, 1)‐circuits. The characterisation uses the well‐known 1‐extension and X‐replacement operations as well as several summation moves to glue together (2, 1)‐circuits over small cutsets.     

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.