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It is shown that every (2p + 1) log 2 (|V (G)|)-edge-connected graph G has a mod (2p + 1)-orientation, and that a (4p + 1)-regular graph G has a mod (2p + 1)-orientation if and only if V (G) has a partition (V + , V −) such that ∀U ⊆ V (G), These extend former results by Da Silva and Dahad on nowhere zero 3-flows of 5-regular graphs, and by Lai and Zhang on… (More)

A graph G is triangularly connected if for every pair of edges e 1 , e 2 ∈ E(G), G has a sequence of 3-cycles C 1 , C 2 , · · · , C l such that e 1 ∈ C 1 , e 2 ∈ C l and such that E(C i) ∩ E(C i+1) = ∅, (1 ≤ i ≤ l − 1). In this paper it is shown that every triangularly connected claw-free graph G with |E(G)| ≥ 3 is vertex pancyclic. This implies the former… (More)

Given a graph G, for an integer c

A graph G is N 2-locally connected if for every vertex v in G, the edges not incident with v but having at least one end adjacent to v in G induce a connected graph. In 1990, Ryjá˘ cek conjectured that every 3-connected N 2-locally connected claw-free graph is hamiltonian. This conjecture is proved in this note.

We investigate graphs G such that the line graph L(G) is hamiltonian connected if and only if L(G) is 3-connected, and prove that if each 3-edge-cut contains an edge lying in a short cycle of G, then L(G) has the above mentioned property. Our result extends Kriesell's recent result in [

Thomassen conjectured that every 4-connected line graph is Hamiltonian. A vertex cut X of G is essential if G − X has at least two non-trivial components. We prove that every 3-connected, essentially 11-connected line graph is Hamiltonian. Using Ryjá˘ cek's line graph closure, it follows that every 3-connected, essentially 11-connected claw-free graph is… (More)

In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: Abstract A sequence d = (d 1 , d 2 ,. .. , d n) is graphic if there is a simple graph… (More)

In 1950s, Tutte introduced the theory of nowhere-zero flows as a tool to investigate the coloring problem of maps, together with his most fascinating conjectures on nowhere-zero flows. These have been extended by Jaeger et al. in 1992 to group connectivity, the nonhomogeneous form of nowhere-zero flows. Let G be a 2-edge-connected undirected graph, A be an… (More)

We prove that every line graph of a 4-edge-connected graph is Z 3-connected. In particular, every line graph of a 4-edge-connected graph has a nowhere zero 3-flow.

Let G be a graph. For u, v ∈ V (G) with distG(u, v) = 2, denote JG(u, v) = {w ∈ NG(u) ∩ NG(v)|NG(w) ⊆ NG(u) ∪ NG(v) ∪ {u, v}}. A graph G is called quasi claw-free if JG(u, v) = ∅ for any u, v ∈ V (G) with distG(u, v) = 2. In 1986, Thomassen conjectured that every 4-connected line graph is hamiltonian. In this paper we show that every 4-connected line graph… (More)