Path factors in cubic graphs

Abstract

Let p be a set of connected graphs. An p -factor of a graph is its spanning subgraph such that each component is isomorphic to one of the members in p . Let P k denote the path of order k. Akiyama and Kano have conjectured that every 3-connected cubic graph of order divisible by 3 has a fP 3 g-factor. Recently, Kaneko gave a necessary and suf®cient condition for a graph to have a fP 3 , P 4 , P 5 g-factor. As a corollary, he proved that every cubic graph has a fP 3 , P 4 , P 5 g-factor. In this paper, we prove that every 2-connected cubic graph of order at least six has a fP k j k ! 6g-factor, and hence has a fP 3 , P 4 g-factor. ß

The clique-transversal number of a{K1,3,K4}-free 4-regular graph

“…Kawarabayashi et al [11] proved that every 2-connected cubic graph G has a 2-factor in which each component is a cycle of length at least four. Theorem 2.5 (Kawarabayashi et al [11]).…”

“…Theorem 2.5 (Kawarabayashi et al [11]). Every 2-connected cubic graph G has a 2-factor in which each component is a cycle of length at least four.…”

An approximation algorithm for maximum -packing in subcubic graphs

“…A partial (positive) answer to this was obtained by Kaneko et al [6] who established that every connected n-vertex claw-free graph having at most two end-blocks (in particular, a 2-connected claw-free graph) has a maximum P 3 -matching of cardinality n/3 . Another result concerning Akiyama and Kano's conjecture was obtained by Kawarabayashi et al [7] who showed that any 2-connected cubic graph of at least 6 vertices has a {P n 6 }-factor, and hence a {P 3 , P 4 }-factor. Clearly, this implies the existence of a P 3 -matching of cardinality at least n/4 in this class of graphs.…”

Path and cycle factors of cubic bipartite graphs

“…It follows immediately from Theorem 4 that every connected cubic bipartite graph G of order at most 16 has a Hamiltonian path since G has a {C n |n ≥ 6}-factor, which consists of at most two components, and a graph consisting of two disjoint cycles and one edge joining them has a Hamiltonian path. It is not mentioned in [4] that the conclusion of Theorem 3 is best possible. However, we can easily find 2-connected cubic graphs having no {C n | n ≥ 5}-factors.…”

“…2 Statement (ii) of Theorem 4 follows immediately from the next Lemma 9 and the statement (i) of Theorem 4. It is shown in [4] that if a 2-connected cubic graph of order at least six has a {C n |n ≥ 4}-factor, then it has a {P n |n ≥ 6}-factor. This statement can be generalized as the following Lemma 9 without changing the proof.…”