Bibliographic Details
| Title: |
A nonplanar critical multigraph that achieves Goldberg's bound. |
| Authors: |
Wang, Chunxiang1 (AUTHOR) wcxiang@ccnu.edu.cn, Wang, Shujing1 (AUTHOR) wang06021@126.com |
| Source: |
Discrete Applied Mathematics. Jun2026, Vol. 386, p381-387. 7p. |
| Subjects: |
Multigraph, Goldberg, Jeffrey, 1965-, Planar graphs, Graph theory, Graph coloring |
| Abstract: |
Let G be a multigraph with maximum degree Δ (G) and odd girth g o (G). Denote by χ ′ (G) the chromatic index of G. Goldberg showed that χ ′ (G) ≤ Δ (G) + 1 + Δ (G) − 2 g o (G) − 1 . For given Δ and odd number g o , let GO (Δ , g o) ≔ { G : Δ (G) = Δ , g o (G) = g o , χ ′ (G) = Δ + 1 + Δ − 2 g o − 1 }. Stiebitz, Scheide, Toft, and Favrholdt in their book conjecture that if G ∈ GO (Δ , g o) , then G contains a ring graph R ∈ GO (Δ , g o) , where ring graph is obtained from a cycle by duplicating some edges. A graph G is critical if χ ′ (H) < χ ′ (G) for any proper subgraph H of G. In this paper, we present a characterization of critical graphs by using near-perfect matchings. Furthermore, we provide some nonplanar critical graphs that achieve Goldberg's bound, which implies that there exists graph G ∈ GO (Δ , g o) , such that for any subgraph H of G with H ∈ GO (Δ , g o) , H is neither a ring graph nor a planar graph. [ABSTRACT FROM AUTHOR] |
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| Database: |
Engineering Source |
