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    Towards the Overfull Conjecture

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    Let GG be a simple graph with maximum degree denoted as Δ(G)\Delta(G). An overfull subgraph HH of GG is a subgraph satisfying the condition ∣E(H)∣>Δ(G)⌊12∣V(H)∣⌋|E(H)| > \Delta(G)\lfloor \frac{1}{2}|V(H)| \rfloor. In 1986, Chetwynd and Hilton proposed the Overfull Conjecture, stating that a graph GG with maximum degree Δ(G)>13∣V(G)∣\Delta(G)> \frac{1}{3}|V(G)| has chromatic index equal to Δ(G)\Delta(G) if and only if it does not contain any overfull subgraph. The Overfull Conjecture has many implications. For example, it implies a polynomial-time algorithm for determining the chromatic index of graphs GG with Δ(G)>13∣V(G)∣\Delta(G) > \frac{1}{3}|V(G)|, and implies several longstanding conjectures in the area of graph edge colorings. In this paper, we make the first improvement towards the conjecture when not imposing a minimum degree condition on the graph: for any 0<ε≤1220<\varepsilon \le \frac{1}{22}, there exists a positive integer n0n_0 such that if GG is a graph on n≥n0n\ge n_0 vertices with Δ(G)≥(1−ε)n\Delta(G) \ge (1-\varepsilon)n, then the Overfull Conjecture holds for GG. The previous best result in this direction, due to Chetwynd and Hilton from 1989, asserts the conjecture for graphs GG with Δ(G)≥∣V(G)∣−3\Delta(G) \ge |V(G)|-3.Comment: arXiv admin note: text overlap with arXiv:2205.08564, arXiv:2105.0528
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