An algorithm on strong edge coloring of \(K_4\)-minor free graphs

Abstract

The strong chromatic index \( \chi’_s(G) \) of a graph \( G \) is the smallest integer \( k \) such that \( G \) has a proper edge \( k \)-coloring with the condition that any two edges at distance at most 2 receive distinct colors. It is known that \( \chi’_s(G) \leq 3\Delta – 2 \) for any \( K_4 \)-minor free graph \( G \) with \( \Delta \geq 3 \). We give a polynomial algorithm in order \( O(|E(G)|(n\Delta^2 + 2n + 14\Delta)) \) to strong color the edges of a \( K_4 \)-minor free graph with \( 3\Delta – 2 \) colors where \( \Delta \geq 3 \).