Minor‐universal planar graphs without accumulation points

We show that the countably infinite union of infinite grids, H say, is minor‐universal in the class of all graphs that can be drawn in the plane without vertex accumulation points, i.e., that H contains every such graph as a minor. Furthermore, we characterize the graphs that occur as minors of the infinite grid by a natural topological condition on their embeddings. © 2000 John Wiley & Sons, Inc. J Graph Theory 36: 1–7, 2001     

Random graphs containing few disjoint excluded minors

The Erd?s‐Pósa theorem (1965) states that in each graph G which contains at most k disjoint cycles, there is a ‘blocking’ set B of at most f(k) vertices such that the graph G – B is acyclic. Robertson and Seymour (1986) give an extension concerning any minor‐closed class of graphs, as long as does not contain all planar graphs: in each graph G which contains at most k disjoint excluded minors for , there is a set B of at most g(k) vertices such that G – B is in . In an earlier paper (Kurauskas and McDiarmid, Combin, Probab Comput 20 (2011) 763–775), we showed that, amongst all graphs on vertex set which contain at most k disjoint cycles, all but an exponentially small proportion contain a blocking set of just k vertices. In the present paper we build on the previous work, and give an extension concerning any minor‐closed graph class with 2‐connected excluded minors, as long as does not contain all fans (here a ‘fan’ is a graph consisting of a path together with a vertex joined to each vertex on the path). We show that amongst all graphs G on which contain at most k disjoint excluded minors for , all but an exponentially small proportion contain a set B of k vertices such that G – B is in . (This is not the case when contains all fans.) For a random graph R_n sampled uniformly from the graphs on with at most k disjoint excluded minors for , we consider also vertex degrees and the uniqueness of small blockers, the clique number and chromatic number, and the probability of being connected. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 240‐268, 2014     

1‐Factorizations of pseudorandom graphs

A 1‐factorization of a graph G is a collection of edge‐disjoint perfect matchings whose union is E(G). In this paper, we prove that for any ?>0, an (n,d,λ)‐graph G admits a 1‐factorization provided that n is even, C₀ ≤ d ≤ n?1 (where C₀=C₀(?) is a constant depending only on ?), and λ ≤ d^1??. In particular, since (as is well known) a typical random d‐regular graph G_n,d is such a graph, we obtain the existence of a 1‐factorization in a typical G_n,d for all C₀ ≤ d ≤ n?1, thereby extending to all possible values of d results obtained by Janson, and independently by Molloy, Robalewska, Robinson, and Wormald for fixed d. Moreover, we also obtain a lower bound for the number of distinct 1‐factorizations of such graphs G, which is better by a factor of 2^nd/2 than the previously best known lower bounds, even in the simplest case where G is the complete graph.     

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     

Additive and hereditary properties of graphs are uniquely factorizable into irreducible factors

A hereditary property of graphs is any class of graphs closed under isomorphism and subgraphs. Let 𝒫₁, 𝒫₂,…, 𝒫_n be hereditary properties of graphs. We say that a graph G has property 𝒫_1°𝒫_2°···_°𝒫_n if the vertex set of G can be partitioned into n sets V₁, V₂,…, V_n such that the subgraph of G induced by V_i belongs to 𝒫_i; i = 1, 2,…, n. A hereditary property is said to be reducible if there exist hereditary properties 𝒫₁ and 𝒫₂ such that ℛ = 𝒫_1°𝒫₂; otherwise it is irreducible. We prove that the factorization of a reducible hereditary property into irreducible factors is unique whenever the property is additive, i.e., it is closed under the disjoint union of graphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 44–53, 2000     

The order of the largest complete minor in a random graph

Let ccl(G) denote the order of the largest complete minor in a graph G (also called the contraction clique number) and let G_n,p denote a random graph on n vertices with edge probability p. Bollobás, Catlin, and Erd?s (Eur J Combin 1 (1980), 195–199) asymptotically determined ccl(G_n,p) when p is a constant. ?uczak, Pittel and Wierman (Trans Am Math Soc 341 (1994) 721–748) gave bounds on ccl(G_n,p) when p is very close to 1/n, i.e. inside the phase transition. We show that for every ε > 0 there exists a constant C such that whenever C/n < p < 1 ‐ ε then asymptotically almost surely ccl(G_n,p) = (1 ± ε)n/ , where b := 1/(1 ‐ p). If p = C/n for a constant C > 1, then ccl(G_n,p) = Θ( ). This extends the results in (Bollobás, Catlin, and P. Erd?s, Eur J Combin 1 (1980), 195–199) and answers a question of Krivelevich and Sudakov (preprint, 2006). © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

On a problem of R. Halin concerning infinite graphs II

For every countable, connected graph A containing no one-way infinite path the following is shown: Let G be an arbitrary graph which contains for every positive integer n a system of n disjoint graphs each isomorphic to a subdivision of A. Then G also contains infinitely many disjoint subgraphs each isomorphic to a subdivision of A. In addition, corrections of errors are given that occur unfortunately in the forerunner of the present paper.     

Edge disjoint Hamilton cycles in sparse random graphs of minimum degree at least k

Let G_n,m,k denote the space of simple graphs with n vertices, m edges, and minimum degree at least k, each graph G being equiprobable. Let G have property A_k, if G contains ⌊(k − 1)/2⌋ edge disjoint Hamilton cycles, and, if k is even, a further edge disjoint matching of size ⌊n/2⌋. We prove that, for k ≥ 3, there is a constant C_k such that if 2m ≥ C_kn then A_k occurs in G_n,m,k with probability tending to 1 as n → ∞. © 2000 John Wiley & Sons, Inc. J. Graph Theory 34: 42–59, 2000     

Embeddings of bipartite graphs

If G is a bipartite graph with bipartition A, B then let G_m,n(A, B) be obtained from G by replacing each vertex a of A by an independent set a₁, …, a_m, each vertex b of B by an independent set b₁,…, b_n, and each edge ab of G by the complete bipartite graph with edges a_ib_j (1 ≤ i ≤ m and 1 ≤ j ≤ n). Whenever G has certain types of spanning forests, then cellular embeddings of G in surfaces S may be lifted to embeddings of G_m,n(A, B) having faces of the same sizes as those of G in S. These results are proved by the technique of “excess-current graphs.” They include new genus embeddings for a large class of bipartite graphs.     

Über eine eigenschaft lokalfiniter,unendlicher bäume

Let A be an arbitrary locally finite, infinite tree and assume that a graph G contains for every positive integer n a system of n disjoint graphs each isomorphic to a subdivision of A. Then G contains infinitely many disjoint subgraphs each isomorphic to a subdivision of A. This sharpens a theorem of Halin [5], who proved the corresponding result for the case that A is a tree in which each vertex has degree not greater than 3.     

Coloring graphs which have equibipartite complements

If G is a graph of order $2n \geq 4$ with an equibipartite complement, then G is Class 1 (i.e., the chromatic index of G is Δ (G)) if and only if G is not the union of two disjoint K_n's with n odd. Similarly if G is a graph of order 2n ≥ 6 whose complement G is equibipartite with bipartition (A, D), and if both G and B, the induced bipartite subgraph of G with bipartition (A, D), have a 1-factor, then G is Type 1 (i.e., the total chromatic number of G is Δ (G) + 1). © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 183–194, 1997     

On the “largeur d'arborescence”

Let la(G) be the invariant introduced by Colin de Verdière [J. Comb. Theory, Ser. B., 74:121–146, 1998], which is defined as the smallest integer n≥0 such that G is isomorphic to a minor of K_n×T, where K_n is a complete graph on n vertices and where T is an arbitrary tree. In this paper, we give an alternative definition of la(G), which is more in terms of the tree‐width of a graph. We give the collection of minimal forbidden minors for the class of graphs G with la(G)≤k, for k=2, 3. We show how this work on la(G) can be used to get a forbidden minor characterization of the graphs with (G)≤3. Here, (G) is another graph parameter introduced in the above cited paper. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 24–52, 2002     

Coloring quasi‐line graphs

A graph G is a quasi‐line graph if for every vertex v, the set of neighbors of v 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. A theorem of Shannon's implies that if G is a line graph, then it can be properly colored using no more than 3/2 ω(G) colors, where ω(G) is the size of the largest clique in G. In this article, we extend this result to all quasi‐line graphs. We also show that this bound is tight. © 2006 Wiley Periodicals, Inc. J Graph Theory     

Recognizing connectedness from vertex‐deleted subgraphs

Let G be a graph of order n. The vertex‐deleted subgraph G ? v, obtained from G by deleting the vertex v and all edges incident to v, is called a card of G. Let H be another graph of order n, disjoint from G. Then the number of common cards of G and H is the maximum number of disjoint pairs (v, w), where v and w are vertices of G and H, respectively, such that G ? v?H ? w. We prove that if G is connected and H is disconnected, then the number of common cards of G and H is at most ?n/2? + 1. Thus, we can recognize the connectedness of a graph from any ?n/2? + 2 of its cards. Moreover, we completely characterize those pairs of graphs that attain the upper bound and show that, with the exception of six pairs of graphs of order at most 7, any pair of graphs that attains the maximum is in one of four infinite families. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:285‐299, 2011     

G-decomposition of Kn, where G has four vertices or less

The complete graph K_n, is said to have a G-decomposition if it is the union of edge disjoint subgraphs each isomorphic to G. The set of values of n for which K_n has a G-decomposition is determined if G has four vertices or less.     

Acyclic Edge Coloring of Triangle‐Free Planar Graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a′(G) ? Δ + 2, where Δ = Δ(G) denotes the maximum degree of the graph. If every induced subgraph H of G satisfies the condition |E(H)| ? 2|V(H)|?1, we say that the graph G satisfies Property A. In this article, we prove that if G satisfies Property A, then a′(G) ? Δ + 3. Triangle‐free planar graphs satisfy Property A. We infer that a′(G) ? Δ + 3, if G is a triangle‐free planar graph. Another class of graph which satisfies Property A is 2‐fold graphs (union of two forests). © 2011 Wiley Periodicals, Inc. J Graph Theory     

Graphs without isometric rays and invariant subgraph properties,I

A ray of a graph G is isometric if every path in R is a shortest path in G. A vertex x of G geodesically dominates a subset A of V(G) if, for every finite S ⊆ V(G − x), there exists an element a of A − {x} such that the interval (set of vertices of all shortest paths) between x and a is disjoint from S. A set A ⊆ V(G) is geodesically closed if it contains all vertices which geodesically dominate A. These geodesically closed sets define a topology, called the geodesic topology, on V(G). We prove that a connected graph G has no isometric rays if and only if the set V(G) endowed with the geodesic topology is compact, or equivalently if and only if the vertex set of every ray in G is geodesically dominated. We prove different invariant subgraph properties for graphs containing no isometric rays. In particular we show that every self-contraction (map which preserves or contracts the edges) of a chordal graph G stabilizes a non-empty finite simplex (complete graph) if and only if G is connected and contains no isometric rays and no infinite simplices. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 99–109, 1998     

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     

Orientation distance graphs

For two nonisomorphic orientations D and D′ of a graph G, the orientation distance d_o(D,D′) between D and D′ is the minimum number of arcs of D whose directions must be reversed to produce an orientation isomorphic to D′. The orientation distance graph 𝒟_o(G) of G has the set 𝒪(G) of pairwise nonisomorphic orientations of G as its vertex set and two vertices D and D′ of 𝒟₀(G) are adjacent if and only if d_o(D,D′) = 1. For a nonempty subset S of 𝒪(G), the orientation distance graph 𝒟₀(S) of S is the induced subgraph 〈S〉 of 𝒟_o(G). A graph H is an orientation distance graph if there exists a graph G and a set S⊆ 𝒪(G) such that 𝒟_o(S) is isomorphic to H. In this case, H is said to be an orientation distance graph with respect to G. This paper deals primarily with orientation distance graphs with respect to paths. For every integer n ≥ 4, it is shown that 𝒟_o(P_n) is Hamiltonian if and only if n is even. Also, the orientation distance graph of a path of odd order is bipartite. Furthermore, every tree is an orientation distance graph with respect to some path, as is every cycle, and for n ≥ 3 the clique number of 𝒟_o(P_n) is 2 if n is odd and is 3 otherwise. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 230–241, 2001