Abstract
We determine (partly by computer search) the chromatic index (edge-chromatic number) of many strongly regular graphs (SRGs), including the SRGs of degree k ≤ 18 and their complements, the Latin square graphs and their complements, and the triangular graphs and their complements. Moreover, using a recent result of Ferber and Jain, we prove that an SRG of even order n, which is not the block graph of a Steiner 2-design or its complement, has chromatic index k, when n is big enough. Except for the Petersen graph, all investigated connected SRGs of even order have chromatic index equal to k, i.e., they are class 1, and we conjecture that this is the case for all connected SRGs of even order.
Highlights
An edge-coloring of a graph G is a coloring of its edges such that intersecting edges have different colors
If G is regular of degree k, G is class 1 if and only if G has an edge coloring such that each color class is a perfect matching
There exist useful spectral conditions for the existence of a perfect matching, and Brouwer and Haemers [5] have shown that every regular graph of even order, degree k and second largest eigenvalue θ2 contains at least ⌊(k − θ2 + 1)/2⌋ edge disjoint perfect matchings
Summary
An edge-coloring of a graph G is a coloring of its edges such that intersecting edges have different colors. There exist useful spectral conditions for the existence of a perfect matching (see [5, 9]), and Brouwer and Haemers [5] have shown that every regular graph of even order, degree k and second largest eigenvalue θ2 contains at least ⌊(k − θ2 + 1)/2⌋ edge disjoint perfect matchings From this it follows that every connected SRG of even order has a perfect matching. By computer, using SageMath [22], we verified that all primitive SRGs of even order and degree k ≤ 18 and their complements are class 1, except for the Petersen graph, which has parameters (10, 3, 0, 1) and edge-chromatic number 4 (see [20, 25] for example). Except for the Petersen graph, every connected SRG of even order is class 1
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
