T-Colorings and T-Edge Spans of Graphs

Abstract

Suppose G is a graph and T is a set of non-negative integers that contains 0. A T-coloring of G is an assignment of a non-negative integer fOxU to each vertex x of G such that j fOxUˇfO yUj B T whenever xy A EOGU. The edge span of a T-coloring f is the maximum value ofj fOxUˇfO yUj over all edges xy , and the T-edge span of a graph G is the minimum value of the edge span of a T-coloring of G. This paper studies the T-edge span of the dth power C d n of the n-cycle Cn for Taf0;1;2; ... ; kˇ 1g. In particular, we find the exact value of the T-edge span of C d n for n10 or 1 (mod da 1), and lower and upper bounds for other cases.