Showing papers by "Hong-Jian Lai published in 2005"
TL;DR: It is proved that if G ∈ C(6,0), then G is supereulerian if and only if G cannot be contracted to K 2,3, K2,5 or K1,3(e), where e ∈ E(K2,3) and K2+, where e stands for a graph obtained from K3,3 by replacing e by a path of length 2.
59 citations
TL;DR: It is shown that using Ryjacek's line graph closure, it follows that every 3-connected, essentially 11-connected claw-free graph is Hamiltonian.
Abstract: 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 nontrivial components. We prove that every 3 -connected, essentially 11 -connected line graph is hamiltonian. Using Ryjacek's line graph closure, it follows that every 3 -connected, essentially 11 -connected claw-free graph is hamiltonian.
22 citations
TL;DR: In this paper, it was shown that every 3-connected N2-locally connected claw-free graph is Hamiltonian, and this conjecture was later proved in this paper.
Abstract: 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 ½ in G induce a connected graph. In 1990, Ryjacek 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: 142146, 2005
11 citations
TL;DR: The group chromatic number of a graph G is defined to be the least positive integer m for which G is A-colorable for any Abelian group A of order ≥m, and is denoted by χg(G).
Abstract: Let G be a graph and A an Abelian group. Denote by F(G, A) the set of all functions from E(G) to A. Denote by D an orientation of E(G). For f ? F(G,A), an (A,f)-coloring of G under the orientation D is a function c : V(G)?A such that for every directed edge uv from u to v, c(u)?c(v) ? f(uv). G is A-colorable under the orientation D if for any function f ? F(G, A), G has an (A, f)-coloring. It is known that A-colorability is independent of the choice of the orientation. The group chromatic number of a graph G is defined to be the least positive integer m for which G is A-colorable for any Abelian group A of order ?m, and is denoted by ? g (G). In this note we will prove the following results. (1) Let H 1 and H 2 be two subgraphs of G such that V(H 1)?V(H 2)=? and V(H 1)?V(H 2)=V(G). Then ? g (G)≤min{max{? g (H 1), max v ? V(H 2) deg(v,G)+1},max{? g (H 2), max u ? V(H 1) deg (u, G) + 1}}. We also show that this bound is best possible. (2) If G is a simple graph without a K 3,3-minor, then ? g (G)≤5.
9 citations
TL;DR: It is shown that if G∈ and κ′(G)≥3, then for every pair of edges e,f∈E(G), G has a trail with initial edge e and final edge f which contains all vertices of G.
Abstract: Suppose that [InlineMediaObject not available: see fulltext.] is the set of connected graphs such that a graph G?[InlineMediaObject not available: see fulltext.] if and only if G satisfies both (F1) if X is an edge cut of G with |X|≤3, then there exists a vertex v of degree |X| such that X consists of all the edges incident with v in G, and (F2) for every v of degree 3, v lies in a k-cycle of G, where 2≤k≤3.
In this paper, we show that if G?[InlineMediaObject not available: see fulltext.] and ??(G)?3, then for every pair of edges e,f?E(G), G has a trail with initial edge e and final edge f which contains all vertices of G. This result extends several former results.
8 citations
01 Jan 2005
4 citations
01 Jan 2005
TL;DR: In this article, it was shown that every 3-connected, essentially 11-connected claw-free graph is hamiltonian, and they also presented two related results concerning hamiltonians.
Abstract: A graph is claw-free if it does not have an induced subgraph isomorphic to a K1,3. In this paper, we proved the every 3-connected, essentially 11-connected claw-free graph is hamiltonian. We also present two related results concerning hamiltonian claw-free graphs.
2 citations
TL;DR: In this paper, it was shown that if a coloopless regular matroid M does not have a minor in {M(K5), M*(k5)], then M admits a nowhere zero 4-flow.
Abstract: Jensen and Toft [8] conjectured that every 2-edge-connected graph without a K5-minor has a nowhere zero 4-flow. Walton and Welsh [19] proved that if a coloopless regular matroid M does not have a minor in {M(K3,3), M*(K5)}, then M admits a nowhere zero 4-flow. In this note, we prove that if a coloopless regular matroid M does not have a minor in {M(K5), M*(K5)}, then M admits a nowhere zero 4-flow. Our result implies the Jensen and Toft conjecture. © 2005 Wiley Periodicals, Inc. J Graph Theory
2 citations
Journal Article•
TL;DR: It is proved that if a 2-edge-connected planar graph G is at most three edges short of having two edge-disjoint spanning trees, then G is supereulerian except a few classes of graphs.
Abstract: We investigate the supereulerian graph problems within planar graphs, and we prove that if a 2-edge-connected planar graph G is at most three edges short of having two edge-disjoint spanning trees, then G is supereulerian except a few classes of graphs. This is applied to show the existence of spanning Eulerian subgraphs in planar graphs with small edge cut conditions. We also determine several extremal bounds for planar graphs to be supereulerian.
1 citations