The total rainbow 2-connection numbers of cubic graphs

Abstract

A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow 2-connected if any two distinct vertices are connected by two internally vertex-disjoint total rainbow paths. The total rainbow 2-connection number of G is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow 2-connected. In this paper, we first study the total rainbow 2-connection numbers of circular ladders and Mobius ladders. Next, we almost determine the total rainbow 2-connection numbers of all small cubic graphs of order 8 or less. © 2018 Utilitas Mathematica Publishing Inc. All rights reserved.