Two-distance vertex-distinguishing index of sparse graphs

Abstract

Abstract The two-distance vertex-distinguishing index χ d 2 ′ ( G ) {\chi }_{d2}^{^{\prime} }\left(G) of graph G G is defined as the smallest integer k k , for which the edges of G G can be properly colored using k k colors. In this way, any pair of vertices at distance of two have distinct sets of colors. The two-distance vertex-distinguishing edge coloring of graphs can be used to solve some network problems. In this article, we used the method of discharging to prove that if G G is a graph with mad ( G ) < 8 3 \left(G)\lt \frac{8}{3} , then χ d 2 ′ ( G ) ≤ max { 7 , Δ + 2 } {\chi }_{d2}^{^{\prime} }\left(G)\le \max \left\{7,\Delta +2\right\} , which improves the result that a graph G G of Δ ( G ) ≥ 4 \Delta \left(G)\ge 4 has χ d 2 ′ ( G ) ≤ Δ ( G ) + 2 {\chi }_{d2}^{^{\prime} }\left(G)\le \Delta \left(G)+2 if mad ( G ) < 5 2 \left(G)\lt \frac{5}{2} , and χ d 2 ′ ( G ) ≤ Δ ( G ) + 3 {\chi }_{d2}^{^{\prime} }\left(G)\le \Delta \left(G)+3 if mad ( G ) < 8 3 \left(G)\lt \frac{8}{3} .