Injective chromatic index of sparse graphs

A k-injective-edge coloring of a graph G is an assignment of colors, i.e. integers in {1,2,& mldr;,k}, to the edges of G such that e(1) and e(3) receive distinct colors for any three consecutive edges e(1), e(2), e(3) of a path or a 3-cycle. The smallest integer k such that G has an injective-edge coloring is called the injective chromatic index of G, denoted by x(i)'(G). In this paper,we consider the injective-edge coloring of sparse graph G with Delta(G)=5, and prove that x(i)'(G)<= 10 (resp., 11, 12, 13) if mad (G)<= 3714 (resp., 39/14, 17/6, 3).

Automated summary

This paper investigates the injective-edge coloring of a sparse graph G, and proves that when mad(G) meets certain conditions, the injective chromatic index x(i)'(G) has a upper bound.