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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.
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A decomposition method on solving the linear arboricity conjecture
Abstract Alinear forestis a disjoint union of path graphs. Thelinear arboricity of a graph, denoted by , is the least number of linear forests into which the graph can be partitioned. Clearly, for any graph of maximum degree . For the upper bound, the long‐standingLinear Arboricity Conjecture(LAC) due to Akiyama, Exoo, and Harary from 1981 asserts that . A graph is apseudoforestif each of its components contains at most one cycle. In this paper, we prove thatthe union of any two pseudoforests of maximum degree up to 3 can be decomposed into three linear forests. Combining it with a recent result of Wdowinski on the minimum number of pseudoforests into which a graph can be decomposed, we prove that the LAC holds for the following simple graph classes: ‐degenerate graphs with maximum degree , all graphs on nonnegative Euler characteristic surfaces provided the maximum degree , and graphs on negative Euler characteristic surfaces provided the maximum degree , as well as graphs with no ‐minor satisfying some conditions on maximum degrees.
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- Award ID(s):
- 2154331
- PAR ID:
- 10468248
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Journal of Graph Theory
- Volume:
- 105
- Issue:
- 2
- ISSN:
- 0364-9024
- Format(s):
- Medium: X Size: p. 315-332
- Size(s):
- p. 315-332
- Sponsoring Org:
- National Science Foundation
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