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In this paper, we prove that the average degree of Δ-critical graphs with Δ=6 is at least . It improves the known bound for the average degree of 6-critical graphs G with |V(G)|>39.  相似文献
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In 1968, Vizing proposed the following conjecture: If G=(V,E) is a Δ-critical graph of order n and size m, then . This conjecture has been verified for the cases of Δ≤5. In this paper, we prove that when Δ=4. It improves the known bound for Δ=4 when n>6.  相似文献
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Let $\mathbf{N}$ be the set of all positive integers. A list assignment of a graph $G$ is a function $L:V\left(G\right)?{2}^{\mathbf{N}}$ that assigns each vertex $v$ a list $L\left(v\right)$ for all $v\in V\left(G\right)$. We say that $G$ is $L$-$\left(2,1\right)$-choosable if there exists a function $?$ such that $?\left(v\right)\in L\left(v\right)$ for all $v\in V\left(G\right)$, $|?\left(u\right)??\left(v\right)|\ge 2$ if $u$ and $v$ are adjacent, and $|?\left(u\right)??\left(v\right)|\ge 1$ if $u$ and $v$ are at distance 2. The list-$L\left(2,1\right)$-labeling number ${\lambda }_{l}\left(G\right)$ of $G$ is the minimum $k$ such that for every list assignment $L=\left\{L\left(v\right):|L\left(v\right)|=k,\phantom{\rule{0.33em}{0ex}}v\in V\left(G\right)\right\}$, $G$ is $L$-$\left(2,1\right)$-choosable. We prove that if $G$ is a planar graph with girth $g\ge 8$ and its maximum degree $\Delta$ is large enough, then ${\lambda }_{l}\left(G\right)\le \Delta +3$. There are graphs with large enough $\Delta$ and $g\ge 8$ having ${\lambda }_{l}\left(G\right)=\Delta +3$.  相似文献
4.
In 1968, Vizing [Uaspekhi Mat Nauk 23 (1968) 117–134; Russian Math Surveys 23 (1968), 125–142] conjectured that for any edge chromatic critical graph with maximum degree , . This conjecture has been verified for . In this article, by applying the discharging method, we prove the conjecture for . © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 149–171, 2009  相似文献
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The 2-step domination problem is to find a minimum vertex set D of a graph such that every vertex of the graph is either in D or at distance two from some vertex of D.In the present paper,by using a labeling method,we provide an O(m) time algorithm to solve the2-step domination problem on block graphs,a superclass of trees.  相似文献
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