STRONG 3-RAINBOW INDEXES OF BAT GRAPHS

Abstract

Let G be a nontrivial connected edge-colored graph of order m≥3, where adjacent edges are allowed to have the same color. A tree T in G is called a rainbow tree if all edges in T have distinct colors. For a subset of vertices S⊆V(G), the Steiner distance sd(S) is defined as the minimum size of a tree in G that contains all vertices in S. Let k be an integer with 2≤k≤m. An edge-coloring of G is called a strong k-rainbow coloring if, for every subset S⊆V(G) with ∣S∣=k, there exists a rainbow tree of size sd(S) containing S. This study employs a theoretical-analytical method using coloring construction and graph structural analysis to determine the strong 3-rainbow index of bat graphs. The results show that the strong 3-rainbow index of the bat graph Bat(n)for n≥3is given by 2 for n=3, n-3, for 4≤n≤6 or n=8 dan n, for 7 or n≥9