ON P4-DECOMPOSITION OF GRAPHS

Abstract

A graph G is decomposable into subgraphs G1, G2, . . . , Gn of G if no Gi (i = 1, 2, . . . , n) has isolated vertices and the edge set E(G) can be partitioned into the subsets E(G1), E(G2), . . . , E(Gn). If Gi ∼= P4 for all i, then G is called P4-decomposable. In this paper, we show the P4-decomposability of some classes of graphs, and prove in particular that a complete r-partite graph is P4-decomposable if and only if its size is a multiple of 3. We also give an example of a 2-connected graph of size 3k which is not P4-decomposable, disproving a conjecture of Chartrand.