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arXiv 1507.07486math.CO
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局部连通性、局部度条件、禁止诱导子图和圈可扩性

Local Connectivity, Local Degree Conditions, some Forbidden Induced Subgraphs, and Cycle Extendability

Christoph Brause, Dieter Rautenbach, Ingo Schiermeyer

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摘要：

Ryjáček猜想局部连通图是弱泛圈图，本文的研究是基于这一猜想。对于阶至少为$3$的连通局部连通图$G$，我们得到如下结果：如果$G$是$(K_1+(K_1\cup K_2))$ -free的，则$G$是弱泛圈的.如果$G$不含$(K_1+(K_1\cup K_2))$，则当且仅当$2δ(G)\geq n(G)$时，$G$可完全循环扩展。如果$G$不含$\{ K_1+K_1+\bar{K}_3,K_1+P_4\}$或不含$\{ K_1+K_1+\bar{K}_3,K_1+(K_1\cup P_3)\}$，则$G$可完全循环延长。如果$G$不同于$K_1+K_1+\bar{K}_3$且不含$\{ K_1+P_4,K_{1,4},K_2+(K_1\cup K_2)\}$，则$G$可完全循环延长。此外，如果$G$是阶至少为$3$的连通图，使得对于$G$中的阶为$3$的每个导出路$vuw$，$$|N_G(u)\cap N_G(v)\cap N_G(w)|>|N_G(u)\setminus (N_G[v]\cup N_G[w])|$$是完全圈可扩的，则$G$是完全圈可扩的，这意味着每个阶至少为$3$的连通局部Ore或局部Dirac图是完全圈可扩的。

英文摘要：

The research in the present paper was motivated by the conjecture of Ryjáček that every locally connected graph is weakly pancyclic. For a connected locally connected graph $G$ of order at least $3$, our results are as follows: If $G$ is $(K_1+(K_1\cup K_2))$-free, then $G$ is weakly pancyclic. If $G$ is $(K_1+(K_1\cup K_2))$-free, then $G$ is fully cycle extendable if and only if $2δ(G)\geq n(G)$. If $G$ is $\{ K_1+K_1+\bar{K}_3,K_1+P_4\}$-free or $\{ K_1+K_1+\bar{K}_3,K_1+(K_1\cup P_3)\}$-free, then $G$ is fully cycle extendable. If $G$ is distinct from $K_1+K_1+\bar{K}_3$ and $\{ K_1+P_4,K_{1,4},K_2+(K_1\cup K_2)\}$-free, then $G$ is fully cycle extendable. Furthermore, if $G$ is a connected graph of order at least $3$ such that $$|N_G(u)\cap N_G(v)\cap N_G(w)|>|N_G(u)\setminus (N_G[v]\cup N_G[w])|$$ for every induced path $vuw$ of order $3$ in $G$, then $G$ is fully cycle extendable, which implies that every connected locally Ore or locally Dirac graph of order at least $3$ is fully cycle extendable.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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