Maximum $Delta$-edge-colorable subgraphs of class II graphs

A graph $G$ is class II, if its chromatic index is at least $\Delta+1$. Let $H$ be a maximum $\Delta$-edge-colorable subgraph of $G$. The paper proves best possible lower bounds for $\frac{|E(H)|}{|E(G)|}$, and structural properties of maximum $\Delta$-edge-colorable subgraphs. It is shown that every set of vertex-disjoint cycles of a class II graph with $\Delta\geq3$ can be extended to a maximum $\Delta$-edge-colorable subgraph. Simple graphs have a maximum $\Delta$-edge-colorable subgraph such that the complement is a matching. Furthermore, a maximum $\Delta$-edge-colorable subgraph of a simple graph is always class I.

💡 Research Summary

The paper investigates the problem of extracting a largest possible subgraph that can be edge‑colored with Δ colors from a class II graph G (i.e., a graph whose chromatic index χ′(G) ≥ Δ + 1, where Δ is the maximum degree). While Vizing’s theorem tells us that any graph belongs to class I (χ′ = Δ) or class II (χ′ = Δ + 1), many class II graphs cannot be completely Δ‑edge‑colored. The authors therefore ask: how many edges can be retained in a Δ‑edge‑colorable subgraph H of G, and what structural properties does such an H possess?

Main contributions

1. Optimal lower bound on the size of H.
   For every class II graph with Δ ≥ 3 the authors prove the existence of a Δ‑edge‑colorable subgraph H satisfying

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