(1,k)-Choosability of Graphs with Edge Lists Containing Arithmetic Progressions

Abstract

In this paper, we give a strengthening of the 1-2-3 conjecture by restricting all edge lists to be arithmetic progressions. We consider list assignments that take every vertex to a single integer and every edge to an arithmetic progression of integers. We prove that for every graph G with such a list assignment and edge lists have length at least 30(3^(2c(G))), then there exists a proper L-total weighting of G.