The hamiltonicity and path t-coloring of Sierpinski-like graphs

A mapping phi from V(G) to {1, 2,..., t) is called a path t-coloring of a graph G if each G[phi(-1)(i)], for 1 <= i <= t, is a linear forest. The vertex linear arboricity of a graph G, denoted by vla(G), is the minimum t for which G has a path t-coloring. Graphs S[n, k] are obtained from the Sierpiriski graphs S(n. k) by contracting all edges that lie in no induced K-k. In this paper, the hamiltonicity and path t-coloring of Sierpiriski-like graphs S(n, k), S+(n, k), S++(n, k) and graphs S[n, k] are studied. In particular, it is obtained that vla(S(n, k)) = vla(S[n, k]) = inverted right perpendiculark/2inverter left perpendicular for k >= 2. Moreover, the numbers of edge disjoint Hamiltonian paths and Hamiltonian cycles in S(n, k), S+(n, k) and S++(n, k) are completely determined, respectively. Crown Copyright (C) 2012 Published by Elsevier B.V. All rights reserved.