On dominating and spanning circuits in graphs

Abstract

An eulerian subgraph of a graph is called a circuit. As shown by Harary and Nash-Williams, the existence of a Hamilton cycle in the line graph L(G) of a graph G is equivalent to the existence of a dominating circuit in G, i.e., a circuit such that every edge of G is incident with a vertex of the circuit. Important progress in the study of the existence of spanning and dominating circuits was made by Catlin, who defined the reduction of a graph G and showed that G has a spanning circuit if and only if the reduction of G has a spanning circuit. We reline Catlin's reduction technique to obtain a result which contains several known and new sufficient conditions for a graph to have a spanning or dominating circuit in terms of degree-sums of adjacent vertices. In particular, the result implies the truth of the following conjecture of Benhocine et al.: If G is a connected simple graph of order n such that every cut edge of G is incident with a vertex of degree 1 and d(u)+d(1;)>2(jn-1) for every edge uu of G, then, for n sufficiently large, L(G) is hamiltonian.0012-365X/94/$07.00 0 1994-Elsevier Science B.V. All rights reserved SSDI 0012-365X(92)00063-A

Eulerian subgraphs in 3‐edge‐connected graphs and Hamiltonian line graphs

“…then for n sufficiently large, either LðGÞ is Hamiltonian, or the Petersen graph is a nontrivial contraction of G. Theorem E improves a previous result in [7] and [15]. The authors in [9] conjectured that the lower bound in Theorem E can be reduced to n=9 À 1, with the conclusion that either LðGÞ is Hamiltonian or G is contractible to the Petersen graph.…”

On subpancyclic line graphs

“…Corollary 4 also proves that the hamiltonian line graphs of those graphs involving edge degrees in [4] and [6] are pancyclic. Here we show one example as follows: Theorem 6.…”

Chvátal–Erdös Type Conditions for Hamiltonicity of Claw-Free Graphs

“…Thus, our definition of G 0 is a special case of Veldman's definition but is an extension of Shao's definition. Following [16], we call G 0 the core of G. Utilizing Theorem 2.2, Veldman [18] and Shao [16] obtained the following. Note that the Petersen graph P is essentially 4-edge-connected with κ (P) = 3.…”

“…In [18], Veldman defined G 0 with W as an independent subset of D 1 (G) ∪ D 2 (G) (not necessary maximum) and denoted G 0 by I W (G). In [16], Shao defined G 0 for (a) (b) (c) (d) Fig.…”