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Abstract Given a multigraph$$G=(V,E)$$, the edge-coloring problem (ECP) is to color the edges ofGwith the minimum number of colors so that no two adjacent edges have the same color. This problem can be naturally formulated as an integer program, and its linear programming relaxation is referred to as the fractional edge-coloring problem (FECP). The optimal value of ECP (resp. FECP) is called the chromatic index (resp. fractional chromatic index) ofG, denoted by$$\chi '(G)$$(resp.$$\chi ^*(G)$$). Let$$\Delta (G)$$be the maximum degree ofGand let$$\Gamma (G)$$be the density ofG, defined by$$\begin{aligned} \Gamma (G)=\max \left\{ \frac{2|E(U)|}{|U|-1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hspace{5.69054pt}\textrm{and} \hspace{5.69054pt}\textrm{odd} \right\} , \end{aligned}$$whereE(U) is the set of all edges ofGwith both ends inU. Clearly,$$\max \{\Delta (G), \, \lceil \Gamma (G) \rceil \}$$is a lower bound for$$\chi '(G)$$. As shown by Seymour,$$\chi ^*(G)=\max \{\Delta (G), \, \Gamma (G)\}$$. In the early 1970s Goldberg and Seymour independently conjectured that$$\chi '(G) \le \max \{\Delta (G)+1, \, \lceil \Gamma (G) \rceil \}$$. Over the past five decades this conjecture, a cornerstone in modern edge-coloring, has been a subject of extensive research, and has stimulated an important body of work. In this paper we present a proof of this conjecture. Our result implies that, first, there are only two possible values for$$\chi '(G)$$, so an analogue to Vizing’s theorem on edge-colorings of simple graphs holds for multigraphs; second, although it isNP-hard in general to determine$$\chi '(G)$$, we can approximate it within one of its true value, and find it exactly in polynomial time when$$\Gamma (G)>\Delta (G)$$; third, every multigraphGsatisfies$$\chi '(G)-\chi ^*(G) \le 1$$, and thus FECP has a fascinating integer rounding property. 

ABSTRACT A subgraph of a graph with maximum degree is ‐overfullif . Clearly, if contains a ‐overfull subgraph, then its chromatic index is . However, the converse is not true, as demonstrated by the Petersen graph. Nevertheless, three families of graphs are conjectured to satisfy the converse statement: (1) graphs with (the Overfull Conjecture of Chetwynd and Hilton), (2) planar graphs (Seymour's Exact Conjecture), and (3) graphs whose subgraph induced on the set of maximum degree vertices is the union of vertex‐disjoint cycles (the Core Conjecture of Hilton and Zhao). Over the past decades, these conjectures have been central to the study of edge coloring in simple graphs. Progress had been slow until recently, when the Core Conjecture was confirmed by the authors in 2024. This breakthrough was achieved by extending Vizing's classical fan technique to two larger families of trees: the pseudo‐multifan and the lollipop. This paper investigates the properties of these two structures, forming part of the theoretical foundation used to prove the Core Conjecture. We anticipate that these developments will provide insights into verifying the Overfull Conjecture for graphs where the subgraph induced by maximum‐degree vertices has relatively small maximum degree. 

Abstract Let be a simple graph. Let and be the maximum degree and the chromatic index of , respectively. We calloverfullif , andcriticalif for every proper subgraph of . Clearly, if is overfull then . Thecoreof , denoted by , is the subgraph of induced by all its maximum degree vertices. We believe that utilizing the core degree condition could be considered as an approach to attack the overfull conjecture. Along this direction, we in this paper show that for any integer , if is critical with and , then is overfull.