Longest paths and cycles in K_1,3‐free graphs

Abstract

In this article w e show that the standard results concerning longest paths and cycles in graphs can be improved for K,,,-free graphs. We obtain as a consequence of these results conditions for the existence of a hamiltonian path and cycle in K,,,-free graphs.There have been a great many results in recent years dealing with graphs that do not contain a copy of K l , , as an induced subgraph. It appears that this class of graphs is better behaved, in many respects, than graphs in general. It has been shown that for such graphs G, (i) If G is connected of even order then G has a I-factor Throughout this paper 6 will denote the minimal degree of a vertex in the graph. Other terminology not defined in this paper will agree with that in Behzad and Chartrand [2].For graphs that are not necessarily hamiltonian there are several results dealing wth longest paths and cycles. Dirac has shown [3] that if G is a connected graph then G has a path of length 26 or a hamiltonian path. Dirac also proved that if G is a 2-connected graph, then G has a cycle of length at least 26 or a hamiltonian cycle. The examples showing that Dirac's results are sharp are not K,,,-free, and if we add K,,3-free to the hypothesis, we obtain slightly larger lower bounds on the longest paths and cycles in graphs. To prove this, we first need a couple of lemmas about maximal cycles and paths in K,,,-free graphs. The proofs of the these lemmas are straightforward and hence are not included.

Generalized degree conditions for graphs with bounded independence number

“…, uk. Since G is connected there is a path from V ( G ) - 5 1, then the removal of 0 or 1 vertices results in two 2-connected components, each of order at least (n/3) + C -f ( r , m) -r -2. In either case, we obtain a hamiltonian subgraph of G of order at least (n/3) + C -f ( r , r n ) -r -2.…”

“…Let G be a graph of order n such that S,(G) 2 ( n / 3 ) + C and P(G) 5 f ( r , r n ) . Assume that G satisfies the hypotheses of (a), (b), or (c).…”

Induced subgraphs with large degrees at end-vertices for hamiltonicity of claw-free graphs

“…Two famous conjectures in the field of hamiltonicity of graphs are Thomassen's conjecture [10] that every 4-connected line graph is hamiltonian and Matthews and Sumner's conjecture [8] that every 4-connected claw-free graph is hamiltonian. Ryjáček proved these two conjectures are equivalent.…”

Hamiltonian Results in K(l,r)-Free Graphs

“…[5] proved that every 2-connected, claw-free graph with 6 > (p -2)/3 is Hamiltonian. In this paper we investigate Hamilton cycles in K 1 1 ,r-free graph with respect to a minimum degree condition.…”