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Average degrees of edge-chromatic critical graphs | | Yan Cao
; Guantao Chen
; Suyun Jiang
; Huiqing Liu
; Fuliang Lu
; | | Date: |
3 Aug 2017 | | Abstract: | Given a graph $G$, denote by $Delta$, $ar{d}$ and $chi^prime$ the
maximum degree, the average degree and the chromatic index of $G$,
respectively. A simple graph $G$ is called {it edge-$Delta$-critical} if
$chi^prime(G)=Delta+1$ and $chi^prime(H)leDelta$ for every proper
subgraph $H$ of $G$. Vizing in 1968 conjectured that if $G$ is
edge-$Delta$-critical, then $ar{d}geq Delta-1+ frac{3}{n}$. We show that
$$ egin{displaystyle} avd ge egin{cases}
0.69241D-0.15658 quad,: mbox{ if } Deltageq 66,
0.69392D-0.20642quad;,mbox{ if } Delta=65, mbox{ and }
0.68706D+0.19815quad! quadmbox{if } 56leq Deltaleq64.
end{cases}
end{displaystyle}
$$
This result improves the best known bound $frac{2}{3}(Delta +2)$ obtained
by Woodall in 2007 for $Delta geq 56$. Additionally, Woodall constructed an
infinite family of graphs showing his result cannot be improved by well-known
Vizing’s Adjacency Lemma and other known edge-coloring techniques. To over come
the barrier, we follow the recently developed recoloring technique of Tashkinov
trees to expand Vizing fans technique to a larger class of trees. | | Source: | arXiv, 1708.1279 | | Services: | Forum | Review | PDF | Favorites | |
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