Article
| Title: | Degree sums of adjacent vertices for traceability of claw-free graphs (English) |
| Author: | Tian, Tao |
| Author: | Xiong, Liming |
| Author: | Chen, Zhi-Hong |
| Author: | Wang, Shipeng |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 72 |
| Issue: | 2 |
| Year: | 2022 |
| Pages: | 313-330 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | The line graph of a graph $G$, denoted by $L(G)$, has $E(G)$ as its vertex set, where two vertices in $L(G)$ are adjacent if and only if the corresponding edges in $G$ have a vertex in common. For a graph $H$, define $\bar {\sigma }_2 (H) = \min \{ d(u) + d(v) \colon uv \in E(H)\}$. Let $H$ be a 2-connected claw-free simple graph of order $n$ with $\delta (H)\geq 3$. We show that, if $\bar {\sigma }_2 (H) \geq \frac 17 (2n -5)$ and $n$ is sufficiently large, then either $H$ is traceable or the Ryjáček's closure ${\rm cl}(H)=L(G)$, where $G$ is an essentially $2$-edge-connected triangle-free graph that can be contracted to one of the two graphs of order 10 which have no spanning trail. Furthermore, if $\bar {\sigma }_2 (H) > \frac 13 (n-6)$ and $n$ is sufficiently large, then $H$ is traceable. The bound $\frac 13 (n-6)$ is sharp. As a byproduct, we prove that there are exactly eight graphs in the family ${\mathcal G}$ of 2-edge-connected simple graphs of order at most 11 that have no spanning trail, an improvement of the result in Z. Niu et al. (2012). (English) |
| Keyword: | traceable graph |
| Keyword: | line graph |
| Keyword: | spanning trail |
| Keyword: | closure |
| MSC: | 05C07 |
| MSC: | 05C38 |
| MSC: | 05C45 |
| idZBL: | Zbl 07547206 |
| idMR: | MR4412761 |
| DOI: | 10.21136/CMJ.2022.0544-19 |
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| Date available: | 2022-04-21T18:58:09Z |
| Last updated: | 2022-09-08 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/150403 |
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