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.     

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     

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     

Tight bounds for the vertices of degree k in minimally k‐connected graphs

For minimally k‐connected graphs on n vertices, Mader proved a tight lower bound for the number of vertices of degree k in dependence on n and k. Oxley observed 1981 that in many cases a considerably better bound can be given if is used as additional parameter, i.e. in dependence on m, n, and k. It was left open to determine whether Oxley's more general bound is best possible. We show that this is not the case, but give a closely related bound that deviates from a variant of Oxley's long‐standing one only for small values of m. We prove that this new bound is best possible. The bound contains Mader's bound as special case.     

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 .     

Minimally (k,k)‐edge‐connected graphs

For an integer l > 1, the l‐edge‐connectivity of a connected graph with at least l vertices is the smallest number of edges whose removal results in a graph with l components. A connected graph G is (k, l)‐edge‐connected if the l‐edge‐connectivity of G is at least k. In this paper, we present a structural characterization of minimally (k, k)‐edge‐connected graphs. As a result, former characterizations of minimally (2, 2)‐edge‐connected graphs in [J of Graph Theory 3 (1979), 15–22] are extended. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 116–131, 2003     

A characterization of weakly four‐connected graphs

A graph G = (V, E) is called weakly four‐connected if G is 4‐edge‐connected and G ? x is 2‐edge‐connected for all x ∈ V. We give sufficient conditions for the existence of ‘splittable’ vertices of degree four in weakly four‐connected graphs. By using these results we prove that every minimally weakly four‐connected graph on at least four vertices contains at least three ‘splittable’ vertices of degree four, which gives rise to an inductive construction of weakly four‐connected graphs. Our results can also be applied in the problem of finding 2‐connected orientations of graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 217–229, 2006     

Proof of Mader's conjecture on k‐critical n‐connected graphs

Mader conjectured that every k‐critical n‐connected noncomplete graph G has 2k + 2 pairwise disjoint fragments. The author in 9 proved that the conjecture holds if the order of G is greater than (k + 2)n. Now we settle this conjecture completely. © 2004 Wiley Periodicals, Inc. J Graph Theory 45: 281–297, 2004     

Subdivisions of large complete bipartite graphs and long induced paths in k‐connected graphs

It is proved that for every positive integers k, r and s there exists an integer n = n(k,r,s) such that every k‐connected graph of order at least n contains either an induced path of length s or a subdivision of the complete bipartite graph K_k,r. © 2004 Wiley Periodicals, Inc. J Graph Theory 45: 270–274, 2004     

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.     

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     

Asymptotic enumeration of sparse 2‐connected graphs

We determine an asymptotic formula for the number of labelled 2‐connected (simple) graphs on n vertices and m edges, provided that m ‐ n →∞ and m = O(nlog n) as n →∞. This is the entire range of m not covered by previous results. The proof involves determining properties of the core and kernel of random graphs with minimum degree at least 2. The case of 2‐edge‐connectedness is treated similarly. We also obtain formulae for the number of 2‐connected graphs with given degree sequence for most (“typical”) sequences. Our main result solves a problem of Wright from 1983. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

All (k;g)‐cages are k‐edge‐connected

A (k;g)‐cage is a k‐regular graph with girth g and with the least possible number of vertices. In this paper, we prove that (k;g)‐cages are k‐edge‐connected if g is even. Earlier, Wang, Xu, and Wang proved that (k;g)‐cages are k‐edge‐connected if g is odd. Combining our results, we conclude that the (k;g)‐cages are k‐edge‐connected. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 219–227, 2005     

Packing k‐edge trees in graphs of restricted vertex degrees

Let denote the set of graphs with each vertex of degree at least r and at most s, v(G) the number of vertices, and τ_k (G) the maximum number of disjoint k‐edge trees in G. In this paper we show that

• (a1) if G ∈ and s ≥ 4, then τ₂(G) ≥ v(G)/(s + 1),
• (a2) if G ∈ and G has no 5‐vertex components, then τ₂(G) ≥ v(G)4,
• (a3) if G ∈ and G has no k‐vertex component, where k ≥ 2 and s ≥ 3, then τ_k(G) ≥ (v(G) ‐k)/(sk ‐ k + 1), and
• (a4) the above bounds are attained for infinitely many connected graphs.

Our proofs provide polynomial time algorithms for finding the corresponding packings in a graph. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 306–324, 2007     

Hamiltonian N2‐locally connected claw‐free graphs

A graph G is N₂‐locally connected if for every vertex ν in G, the edges not incident with ν but having at least one end adjacent to ν in G induce a connected graph. In 1990, Ryjá?ek conjectured that every 3‐connected N₂‐locally connected claw‐free graph is Hamiltonian. This conjecture is proved in this note. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 142–146, 2005     

Highly edge‐connected detachments of graphs and digraphs

Let G = (V,E) be a graph or digraph and r : V → Z₊. An r‐detachment of G is a graph H obtained by ‘splitting’ each vertex ν ∈ V into r(ν) vertices. The vertices ν₁,…,ν_r(ν) obtained by splitting ν are called the pieces of ν in H. Every edge uν ∈ E corresponds to an edge of H connecting some piece of u to some piece of ν. Crispin Nash‐Williams 9 gave necessary and sufficient conditions for a graph to have a k‐edge‐connected r‐detachment. He also solved the version where the degrees of all the pieces are specified. In this paper, we solve the same problems for directed graphs. We also give a simple and self‐contained new proof for the undirected result. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 67–77, 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