Connectivity keeping paths in k‐connected graphs

A result of G. Chartrand, A. Kaugars, and D. R. Lick [Proc Amer Math Soc 32 (1972), 63–68] says that every finite, k‐connected graph G of minimum degree at least ?3k/2? contains a vertex x such that G?x is still k‐connected. We generalize this result by proving that every finite, k‐connected graph G of minimum degree at least ?3k/2?+m?1 for a positive integer m contains a path P of length m?1 such that G?V(P) is still k‐connected. This has been conjectured in a weaker form by S. Fujita and K. Kawarabayashi [J Combin Theory Ser B 98 (2008), 805–811]. © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 61–69, 2010.     

Connectivity keeping trees in k‐connected graphs

We show that one can choose the minimum degree of a k‐connected graph G large enough (independent of the vertex number of G) such that G contains a copy T of a prescribed tree with the property that G ? V(T) remains k‐connected. This was conjectured in [W. Mader, J Graph Theory 65 (2010), 61–69]. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 324–329, 2012     

On random k‐out subgraphs of large graphs

We consider random subgraphs of a fixed graph with large minimum degree. We fix a positive integer k and let G_k be the random subgraph where each independently chooses k random neighbors, making kn edges in all. When the minimum degree then G_k is k‐connected w.h.p. for ; Hamiltonian for k sufficiently large. When , then G_k has a cycle of length for . By w.h.p. we mean that the probability of non‐occurrence can be bounded by a function (or ) where . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 143–157, 2017     

Graph limits of random graphs from a subset of connected k‐trees

For any set Ω of non‐negative integers such that , we consider a random Ω‐k‐tree G_n,k that is uniformly selected from all connected k‐trees of (n + k) vertices such that the number of (k + 1)‐cliques that contain any fixed k‐clique belongs to Ω. We prove that G_n,k, scaled by where H_k is the kth harmonic number and σ_Ω > 0, converges to the continuum random tree . Furthermore, we prove local convergence of the random Ω‐k‐tree to an infinite but locally finite random Ω‐k‐tree G_∞,k.     

Circuits through prescribed vertices in k‐connected k‐regular graphs

We show that every set of vertices in a k‐connected k‐regular graph belongs to some circuit. © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 145–163, 2002     

Density of the circular chromatic numbers of series‐parallel graphs

In this paper we show that every simple cubic graph on n vertices has a set of at least ? n/4 ? disjoint 2‐edge paths and that this bound is sharp. Our proof provides a polynomial time algorithm for finding such a set in a simple cubic graph. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 57–79, 2003     

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     

A note on k‐shortest paths problem

It is well‐known that in a directed graph, if deleting any edge will not affect the shortest distance between two specific vertices s and t, then there are two edge‐disjoint paths from s to t and both of them are shortest paths. In this article, we generalize this to shortest k edge‐disjoint s‐t paths for any positive integer k. © 2010 Wiley Periodicals, Inc. J Graph Theory 67: 34‐37, 2011     

k‐Kings in k‐Quasitransitive Digraphs

Let D be a digraph with vertex set and arc set . A vertex x is a k‐king of D, if for every , there is an ‐path of length at most k. A subset N of is k‐independent if for every pair of vertices , we have and ; it is l‐absorbent if for every there exists such that . A ‐kernel of D is a k‐independent and l‐absorbent subset of . A k‐kernel is a ‐kernel. A digraph D is k‐quasitransitive, if for any path of length k, x₀ and are adjacent. In this article, we will prove that a k‐quasitransitive digraph with has a k‐king if and only if it has a unique initial strong component and the unique initial strong component is not isomorphic to an extended ‐cycle where each has at least two vertices. Using this fact, we show that every strong k‐quasitransitive digraph has a ‐kernel.     

Contractible subgraphs in k‐connected graphs

For a graph G we define a graph T(G) whose vertices are the triangles in G and two vertices of T(G) are adjacent if their corresponding triangles in G share an edge. Kawarabayashi showed that if G is a k‐connected graph and T(G) contains no edge, then G admits a k‐contractible clique of size at most 3, generalizing an earlier result of Thomassen. In this paper, we further generalize Kawarabayashi's result by showing that if G is k‐connected and the maximum degree of T(G) is at most 1, then G admits a k‐contractible clique of size at most 3 or there exist independent edges e and f of G such that e and f are contained in triangles sharing an edge and G/e/f is k‐connected. © 2006 Wiley Periodicals, Inc. J Graph Theory 55: 121–136, 2007     

Large classes of infinite k‐cop‐win graphs

While finite cop‐win finite graphs possess a good structural characterization, none is known for infinite cop‐win graphs. As evidence that such a characterization might not exist, we provide as large as possible classes of infinite graphs with finite cop number. More precisely, for each infinite cardinal κ and each positive integer k, we construct 2^κ non‐isomorphic k‐cop‐win graphs satisfying additional properties such as vertex‐transitivity, or having universal endomorphism monoid and automorphism group. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 334–342, 2010     

Nowhere‐zero 3‐flows in locally connected graphs

Let G be a graph. For each vertex v ∈V(G), N_v denotes the subgraph induces by the vertices adjacent to v in G. The graph G is locally k‐edge‐connected if for each vertex v ∈V(G), N_v is k‐edge‐connected. In this paper we study the existence of nowhere‐zero 3‐flows in locally k‐edge‐connected graphs. In particular, we show that every 2‐edge‐connected, locally 3‐edge‐connected graph admits a nowhere‐zero 3‐flow. This result is best possible in the sense that there exists an infinite family of 2‐edge‐connected, locally 2‐edge‐connected graphs each of which does not have a 3‐NZF. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 211–219, 2003     

Nowhere‐zero 3‐flow and ‐connectedness in graphs with four edge‐disjoint spanning trees

Given a zero‐sum function with , an orientation D of G with in for every vertex is called a β‐orientation. A graph G is ‐connected if G admits a β‐orientation for every zero‐sum function β. Jaeger et al. conjectured that every 5‐edge‐connected graph is ‐connected. A graph is ‐extendable at vertex v if any preorientation at v can be extended to a β‐orientation of G for any zero‐sum function β. We observe that if every 5‐edge‐connected essentially 6‐edge‐connected graph is ‐extendable at any degree five vertex, then the above‐mentioned conjecture by Jaeger et al. holds as well. Furthermore, applying the partial flow extension method of Thomassen and of Lovász et al., we prove that every graph with at least four edge‐disjoint spanning trees is ‐connected. Consequently, every 5‐edge‐connected essentially 23‐edge‐connected graph is ‐extendable at any degree five vertex.     

Ancestors and descendants in evolving k‐tree models

We consider several random graph models based on k‐trees, which can be generated by applying the probabilistic growth rules “uniform attachment”, “preferential attachment”, or a “saturation”‐rule, respectively, but which also can be described in a combinatorial way. For all of these models we study the number of ancestors and the number of descendants of nodes in the graph by carrying out a precise analysis which leads to exact and limiting distributional results. © 2014 Wiley Periodicals, Inc. Random Struct. Alg. 44, 465–489, 2014     

Long cycles passing through vertices in k‐connected graphs

In this article, we consider the following problem proposed by Locke and Zhang in 1991: Let G be a k‐connected graph with minimum degree d and X a set of m vertices on a cycle of G. For which values of m and k, with , must G have a cycle of length at least passing through X? Fujisawa and Yamashita solved this problem for the case and in 2008. We provide an affirmative answer to this problem for the case of and .     

Game chromatic index of k‐degenerate graphs

We consider the following edge coloring game on a graph G. Given t distinct colors, two players Alice and Bob, with Alice moving first, alternately select an uncolored edge e of G and assign it a color different from the colors of edges adjacent to e. Bob wins if, at any stage of the game, there is an uncolored edge adjacent to colored edges in all t colors; otherwise Alice wins. Note that when Alice wins, all edges of G are properly colored. The game chromatic index of a graph G is the minimum number of colors for which Alice has a winning strategy. In this paper, we study the edge coloring game on k‐degenerate graphs. We prove that the game chromatic index of a k‐degenerate graph is at most Δ + 3k − 1, where Δ is the maximum vertex degree of the graph. We also show that the game chromatic index of a forest of maximum degree 3 is at most 4 when the forest contains an odd number of edges. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 144–155, 2001     

On the structure of graphs with a unique k‐factor

In this paper we study the structure of graphs with a unique k‐factor. Our results imply a conjecture of Hendry on the maximal number m (n,k) of edges in a graph G of order n with a unique k‐factor: For we prove and construct all corresponding extremal graphs. For we prove . For n = 2kl, l ∈ ℕ, this bound is sharp, and we prove that the corresponding extremal graph is unique up to isomorphism. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 227–243, 2000