“Less” Strong Chromatic Indices and the (7, 4)-Conjecture

Abstract

A proper edge coloring of a graph 𝐺 is strong if the union of any two color classes does not contain a path with three edges (i.e. the color classes are induced matchings ). The strong chromatic index 𝑞(𝐺) is the smallest number of colors needed for a strong coloring of 𝐺. One form of the famous (6, 3)-theorem of Ruzsa and Szemerédi (solving the (6, 3)-conjecture of Brown–Erdős–Sós) states that 𝑞(𝐺) cannot be linear in 𝑛 for a graph 𝐺 with 𝑛 vertices and 𝑐𝑛 2 edges. Here we study two refinements of 𝑞(𝐺) arising from the analogous (7, 4)-conjecture. The first is 𝑞 𝐴 (𝐺), the smallest number of colors needed for a proper edge coloring of 𝐺 such that the union of any two color classes does not contain a path or cycle with four edges, we call it an A-coloring . The second is 𝑞 𝐵 (𝐺), the smallest number of colors needed for a proper edge coloring of 𝐺 such that all four-cycles are colored with four different colors, we call it a B-coloring . These notions lead to two stronger and one equivalent form of the (7, 4)-conjecture in terms of 𝑞 𝐴 (𝐺), 𝑞 𝐵 (𝐺) where 𝐺 is a balanced bipartite graph. Since these are questions about graphs, perhaps they will be easier to handle than the original special (7, 4)-conjecture. In order to understand the behavior of 𝑞 𝐴(𝐺) and 𝑞 𝐵(𝐺), we study these parameters for some graphs. We note that 𝑞 𝐴 (𝐺) has already been extensively studied from various motivations. However, as far as we know the behavior of 𝑞 𝐵 (𝐺) is studied here for the first time.