The inclusion-free edge-colorings of (3,Δ)-bipartite graphs

Abstract

An inclusion-free edge-coloring of a graph G with δ ( G ) ≥ 2 is a proper edge-coloring in which the color set of a vertex u is not contained in the color set of a neighbor of u . The least number of colors required in an inclusion-free edge-coloring is the inclusion chromatic index , denoted by χ ⊂ ′ ( G ) . In the paper, we show that if G is a bipartite graph with bipartition ( A , B ) where Δ ( A ) ≤ 3 and δ ( G ) ≥ 2 , then χ ⊂ ′ ( G ) ≤ 2 Δ + 1 .