The Λ-Number OF THE CARTESIAN PRODUCT OF a COMPLETE GRAPH AND a CYCLE

Abstract

An L(j, k)-labeling of a graph G is a vertex labeling such that the difference of the labels of any adjacent vertices is at least j and that of any vertices of distance two is at least k for given j and k. The minimum span of all L(2, 1)-labelings of G is called the λ-number of G and is denoted by λ(G). In this paper, we find a lower bound of the λ-number of the Cartesian product K m C n of the complete graph K m of order m and the cycle C n of order n. In fact, we show that when n ≥ 3, λ(K 4 C n) ≥ 7 and the equality holds if and only if n is a multiple of 8. Moreover when m ≥ 5, λ(K m C n) ≥ 2m − 1 and the equality holds if and only if n is even.

On irreducible no-hole L(2, 1)-coloring of Cartesian product of trees with paths

“…The L(2, 1)-coloring of the Cartesian product of cycles were studied by Jha et al [6], and Schwarz and Troxell [17]. Then Kim et al [8] studied the L(2, 1)-coloring of the Cartesian product of a complete graph and a cycle.…”