Fair reception and Vizing's conjecture

In this paper we introduce the concept of fair reception of a graph which is related to its domination number. We prove that all graphs G with a fair reception of size γ(G) satisfy Vizing's conjecture on the domination number of Cartesian product graphs, by which we extend the well‐known result of Barcalkin and German concerning decomposable graphs. Combining our concept with a result of Aharoni, Berger and Ziv, we obtain an alternative proof of the theorem of Aharoni and Szabó that chordal graphs satisfy Vizing's conjecture. A new infinite family of graphs that satisfy Vizing's conjecture is also presented. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 45‐54, 2009     

Adjacent strong edge colorings and total colorings of regular graphs

It is conjectured that χas(G) = χt(G) for every k-regular graph G with no C5 component (k 2). This conjecture is shown to be true for many classes of graphs, including: graphs of type 1; 2-regular, 3-regular and (|V (G)| - 2)-regular graphs; bipartite graphs; balanced complete multipartite graphs; k-cubes; and joins of two matchings or cycles.     

Total chromatic number of complete r-partite graphs

Rosenfeld (1971) proved that the Total Colouring Conjecture holds for balanced complete r-partite graphs. Bermond (1974) determined the exact total chromatic number of every balanced complete r-partite graph. Rosenfeld's result had been generalized recently to complete r-partite graphs by Yap (1989). The main result of this paper is to prove that the total chromatic number of every complete r-partite graph G of odd order is Δ (G) + 1. This result gives a partial generalization of Bermond's theorem.     

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     

Chordal Bipartite Graphs with High Boxicity

The boxicity of a graph G is defined as the minimum integer k such that G is an intersection graph of axis-parallel k-dimensional boxes. Chordal bipartite graphs are bipartite graphs that do not contain an induced cycle of length greater than 4. It was conjectured by Otachi, Okamoto and Yamazaki that chordal bipartite graphs have boxicity at most 2. We disprove this conjecture by exhibiting an infinite family of chordal bipartite graphs that have unbounded boxicity.     

On zero-sum -flows of graphs

For a graph G, a zero-sum flow is an assignment of non-zero real numbers on the edges of G such that the total sum of all edges incident with any vertex of G is zero. A zero-sum k-flow for a graph G is a zero-sum flow with labels from the set {±1,…,±(k-1)}. In this paper for a graph G, a necessary and sufficient condition for the existence of zero-sum flow is given. We conjecture that if G is a graph with a zero-sum flow, then G has a zero-sum 6-flow. It is shown that the conjecture is true for 2-edge connected bipartite graphs, and every r-regular graph with r even, r>2, or r=3.     

Nowhere‐zero 3‐flows in products of graphs

A graph G is an odd‐circuit tree if every block of G is an odd length circuit. It is proved in this paper that the product of every pair of graphs G and H admits a nowhere‐zero 3‐flow unless G is an odd‐circuit tree and H has a bridge. This theorem is a partial result to the Tutte's 3‐flow conjecture and generalizes a result by Imrich and Skrekovski [7] that the product of two bipartite graphs admits a nowhere‐zero 3‐flow. A byproduct of this theorem is that every bridgeless Cayley graph G = Cay(Γ,S) on an abelian group Γ with a minimal generating set S admits a nowhere‐zero 3‐flow except for odd prisms. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Largest bipartite subgraphs in triangle-free graphs with maximum degree three

Let G be a triangle-free, loopless graph with maximum degree three. We display a polynomial algorithm which returns a bipartite subgraph of G containing at least ? of the edges of G. Furthermore, we characterize the dodecahedron and the Petersen graph as the only 3-regular, triangle-free, loopless, connected graphs for which no bipartite subgraph has more than this proportion.     

Strong chromatic index of subset graphs

The strong chromatic index of a graph G, denoted sq(G), is the minimum number of parts needed to partition the edges of G into induced matchings. For 0 ≤ k ≤ l ≤ m, the subset graph S_m(k, l) is a bipartite graph whose vertices are the k- and l-subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. We show that and that this number satisfies the strong chromatic index conjecture by Brualdi and Quinn for bipartite graphs. Further, we demonstrate that the conjecture is also valid for a more general family of bipartite graphs. © 1997 John Wiley & Sons, Inc.     

Hadwiger's conjecture for quasi‐line graphs

A graph G is a quasi‐line graph if for every vertex v ∈ V(G), the set of neighbors of v in G can be expressed as the union of two cliques. The class of quasi‐line graphs is a proper superset of the class of line graphs. Hadwiger's conjecture states that if a graph G is not t‐colorable then it contains K_{t + 1} as a minor. This conjecture has been proved for line graphs by Reed and Seymour. We extend their result to all quasi‐line graphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 17–33, 2008     

NP completeness of the edge precoloring extension problem on bipartite graphs

We show that the following problem is NP complete: Let G be a cubic bipartite graph and f be a precoloring of a subset of edges of G using at most three colors. Can f be extended to a proper edge 3‐coloring of the entire graph G? This result provides a natural counterpart to classical Holyer's result on edge 3‐colorability of cubic graphs and a strengthening of results on precoloring extension of perfect graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 156–160, 2003     

Decomposing Oriented Graphs into Six Locally Irregular Oriented Graphs

An undirected graph G is locally irregular if every two of its adjacent vertices have distinct degrees. We say that G is decomposable into k locally irregular graphs if there exists a partition \(E_1 \cup E_2 \cup \cdots \cup E_k\) of the edge set E(G) such that each \(E_i\) induces a locally irregular graph. It was recently conjectured by Baudon et al. that every undirected graph admits a decomposition into at most three locally irregular graphs, except for a well-characterized set of indecomposable graphs. We herein consider an oriented version of this conjecture. Namely, can every oriented graph be decomposed into at most three locally irregular oriented graphs, i.e. whose adjacent vertices have distinct outdegrees? We start by supporting this conjecture by verifying it for several classes of oriented graphs. We then prove a weaker version of this conjecture. Namely, we prove that every oriented graph can be decomposed into at most six locally irregular oriented graphs. We finally prove that even if our conjecture were true, it would remain NP-complete to decide whether an oriented graph is decomposable into at most two locally irregular oriented graphs.     

Every 3‐connected claw‐free Z8‐free graph is Hamiltonian

A biclique of a graph G is a maximal induced complete bipartite subgraph of G. Given a graph G, the biclique matrix of G is a {0,1,?1} matrix having one row for each biclique and one column for each vertex of G, and such that a pair of 1, ?1 entries in a same row corresponds exactly to adjacent vertices in the corresponding biclique. We describe a characterization of biclique matrices, in similar terms as those employed in Gilmore's characterization of clique matrices. On the other hand, the biclique graph of a graph is the intersection graph of the bicliques of G. Using the concept of biclique matrices, we describe a Krausz‐type characterization of biclique graphs. Finally, we show that every induced P₃ of a biclique graph must be included in a diamond or in a 3‐fan and we also characterize biclique graphs of bipartite graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 1–16, 2010     

Maximum k-colorable subgraphs

A lower bound is established on the number of edges in a maximum k-colorable subgraph of a loopless graph G. For the special case of 3-regular graphs, lower bounds are also determined on the maximum number of edges in a bipartite subgraph whose color classes are of equal size.     

An Approximate Version of Sidorenko’s Conjecture

A beautiful conjecture of Erdős-Simonovits and Sidorenko states that, if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. This conjecture also has an equivalent analytic form and has connections to a broad range of topics, such as matrix theory, Markov chains, graph limits, and quasirandomness. Here we prove the conjecture if H has a vertex complete to the other part, and deduce an approximate version of the conjecture for all H. Furthermore, for a large class of bipartite graphs, we prove a stronger stability result which answers a question of Chung, Graham, and Wilson on quasirandomness for these graphs.     

Det‐extremal cubic bipartite graphs

Let G be a connected k–regular bipartite graph with bipartition V(G) = X ∪ Y and adjacency matrix A. We say G is det‐extremal if per (A) = |det(A)|. Det–extremal k–regular bipartite graphs exist only for k = 2 or 3. McCuaig has characterized the det‐extremal 3‐connected cubic bipartite graphs. We extend McCuaig's result by determining the structure of det‐extremal cubic bipartite graphs of connectivity two. We use our results to determine which numbers can occur as orders of det‐extremal connected cubic bipartite graphs, thus solving a problem due to H. Gropp. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 50–64, 2003     

On a conjecture of Murty and Simon on diameter 2-critical graphs

A graph G is diameter 2-critical if its diameter is two, and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter 2-critical graph of order n is at most n²/4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. We use an association with total domination to prove the conjecture for the graphs whose complements have diameter three.     

A proof of a conjecture on diameter 2-critical graphs whose complements are claw-free

A graph G is diameter 2-critical if its diameter is 2, and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter 2-critical graph of order n is at most n²/4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. We use an important association with total domination to prove the conjecture for the graphs whose complements are claw-free.     

Transversal hypergraphs to perfect matchings in bipartite graphs: Characterization and generation algorithms

A minimal blocker in a bipartite graph G is a minimal set of edges the removal of which leaves no perfect matching in G. We give an explicit characterization of the minimal blockers of a bipartite graph G. This result allows us to obtain a polynomial delay algorithm for finding all minimal blockers of a given bipartite graph. Equivalently, we obtain a polynomial delay algorithm for listing the anti‐vertices of the perfect matching polytope of G. We also provide generation algorithms for other related problems, including d‐factors in bipartite graphs, and perfect 2‐matchings in general graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 209–232, 2006     

Meyniel's conjecture holds for random d‐regular graphs

In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C, the cop number of every connected graph G is at most . In a separate paper, we showed that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph. The result was obtained by showing that the conjecture holds for a general class of graphs with some specific expansion‐type properties. In this paper, this deterministic result is used to show that the conjecture holds asymptotically almost surely for random d‐regular graphs when d = d(n) ≥ 3.