| Title:
|
On the order of certain close to regular graphs without a matching of given size (English) |
| Author:
|
Klinkenberg, Sabine |
| Author:
|
Volkmann, Lutz |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
57 |
| Issue:
|
3 |
| Year:
|
2007 |
| Pages:
|
907-918 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A graph $G$ is a $\lbrace d,d+k\rbrace $-graph, if one vertex has degree $d+k$ and the remaining vertices of $G$ have degree $d$. In the special case of $k=0$, the graph $G$ is $d$-regular. Let $k,p\ge 0$ and $d,n\ge 1$ be integers such that $n$ and $p$ are of the same parity. If $G$ is a connected $\lbrace d,d+k\rbrace $-graph of order $n$ without a matching $M$ of size $2|M|=n-p$, then we show in this paper the following: If $d=2$, then $k\ge 2(p+2)$ and (i) $n\ge k+p+6$. If $d\ge 3$ is odd and $t$ an integer with $1\le t\le p+2$, then (ii) $n\ge d+k+1$ for $k\ge d(p+2)$, (iii) $n\ge d(p+3)+2t+1$ for $d(p+2-t)+t\le k\le d(p+3-t)+t-3$, (iv) $n\ge d(p+3)+2p+7$ for $k\le p$. If $d\ge 4$ is even, then (v) $n\ge d+k+2-\eta $ for $k\ge d(p+3)+p+4+\eta $, (vi) $n\ge d+k+p+2-2t=d(p+4)+p+6$ for $k=d(p+3)+4+2t$ and $p\ge 1$, (vii) $n\ge d+k+p+4$ for $d(p+2)\le k\le d(p+3)+2$, (viii) $n\ge d(p+3)+p+4$ for $k\le d(p+2)-2$, where $0\le t\le \frac{1}{2}{p}-1$ and $\eta =0$ for even $p$ and $0\le t\le \frac{1}{2}(p-1)$ and $\eta =1$ for odd $p$. The special case $k=p=0$ of this result was done by Wallis [6] in 1981, and the case $p=0$ was proved by Caccetta and Mardiyono [2] in 1994. Examples show that the given bounds (i)–(viii) are best possible. (English) |
| Keyword:
|
matching |
| Keyword:
|
maximum matching |
| Keyword:
|
close to regular graph |
| MSC:
|
05C07 |
| MSC:
|
05C70 |
| idZBL:
|
Zbl 1174.05101 |
| idMR:
|
MR2356929 |
| . |
| Date available:
|
2009-09-24T11:50:28Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128215 |
| . |
| Reference:
|
[1] C. Berge: Sur le couplage maximum d’un graphe.C. R. Acad. Sci. Paris 247 (1958), 258–259. (French) Zbl 0086.16301, MR 0100850 |
| Reference:
|
[2] L. Caccetta, S. Mardiyono: On the existence of almost-regular-graphs without one-factors.Australas. J. Comb. 9 (1994), 243–260. MR 1271205 |
| Reference:
|
[3] G. Chartrand, L. Lesniak: Graphs and Digraphs, 3rd Edition.Chapman and Hall, London, 1996. MR 1408678 |
| Reference:
|
[4] W. T. Tutte: The factorization of linear graphs.J. Lond. Math. Soc. 22 (1947), 107–111. Zbl 0029.23301, MR 0023048, 10.1112/jlms/s1-22.2.107 |
| Reference:
|
[5] L. Volkmann: Foundations of Graph Theory.Springer-Verlag, Wien-New York, 1996. (German) MR 1392955 |
| Reference:
|
[6] W. D. Wallis: The smallest regular graphs without one-factors.Ars Comb. 11 (1981), 295–300. Zbl 0468.05042, MR 0629881 |
| . |
