Extremal Problems on Detectable Colorings of Connected Graphs With Cycle Rank 2

Abstract

Let G be a connected graph of order n ≥ 3 and let c: E(G) →) {1, 2,…,k} be a coloring of the edges of G (where adjacent edges may be colored the same). For each vertex v of G, the color code of v with respect to c is the k -tuple c(v) = (a1, a2, …, ak), where ai is the number of edges incident with v that are colored i (1 ≤ i ≤ ki). The coloring c is detectable if distinct vertices have distinct color codes. The detection number det(G) of G is the minimum positive integer k for which G has a detectable k -coloring. A connected graph of order n ≥ 4 and size m is said to have cycle rank 2 if m = n + 1. For each integer n ≥ 4, let D2 (n) be the maximum detection number among all connected graphs of order n with cycle rank 2 and d2(n) the minimum detection number among all connected graphs of order n with cycle rank 2. The numbers D2(n) and d2(n) are determined for all integers n ≥ 4. Furthermore, for integers k ≥ 2 and n ≥ 4, there exists a connected graph G of order n having cycle rank 2 and det(G) = k if and only if d2(n) ≤ k ≤ D2(n).