(2, l)-Total labeling of cube connected cycle
Abstract
A (p, l)-total labeling of a graph G is an assignment of integers to V(G) U E(G) such that any two adjacent vertices of G receive distinct integers, any two adjacent edges of G receive distinct integers, and a vertex and an edge incident receive integers that differ by at least p in absolute value. The span of a (p, l)-total labeling is the maximum difference between two labels. The minimum of span of all possible (p, 1)- total labeling of G is called the (p, l)-total number and is denoted by Ap (G). The well-known Havet and Yu conjecture [17] states that, for any connected graph G with A(G) < 3 and G KA} AJ∗ (G) < 5. In this paper, we determine the (2, l)-total number of a cube connected cycle of order k, where the cube connected cycle of order k is the cubic graph obtained from hypercube of dimension kf Qk, by appropriately replacing every vertex of Qk by a cycle of order k and is denoted by CCGk. More precisely, we show that A2 (CCCk) = 5, for any k > 3. This result supports the Havet and Yu conjecture [17]. © 2018 Utilitas Mathematica Publishing Incorporated. All rights reserved.
