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Abstract Let be a simple graph and be the chromatic index of . We call a ‐critical graphif for every edge of , where is maximum degree of . Let be an edge of ‐critical graph and be an (proper) edge ‐coloring of . Ane‐fanis a sequence of alternating vertices and distinct edges such that edge is incident with or , is another endvertex of and is missing at a vertex before for each with . In this paper, we prove that if , where and denote the degrees of vertices and , respectively, then colors missing at different vertices of are distinct. Clearly, a Vizing fan is an ‐fan with the restricting that all edges being incident with one fixed endvertex of edge . This result gives a common generalization of several recently developed new results on multifan, double fan, Kierstead path of four vertices, and broom. By treating some colors of edges incident with vertices of low degrees as missing colors, Kostochka and Stiebitz introduced ‐fan. In this paper, we also generalize the ‐fan from centered at one vertex to one edge.
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On random irregular subgraphs
Abstract Let be a ‐regular graph on vertices. Frieze, Gould, Karoński, and Pfender began the study of the following random spanning subgraph model . Assign independently to each vertex of a uniform random number , and an edge of is an edge of if and only if . Addressing a problem of Alon and Wei, we prove that if , then with high probability, for each nonnegative integer , there are vertices of degree in .
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- PAR ID:
- 10644755
- Publisher / Repository:
- Wiley
- Date Published:
- Journal Name:
- Random Structures & Algorithms
- Volume:
- 64
- Issue:
- 4
- ISSN:
- 1042-9832
- Page Range / eLocation ID:
- 899 to 917
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1002/rsa.21204
