On the order of certain close to regular graphs without a matching of given size

Title:

On the order of certain close to regular graphs without a matching of given size

Creator:

Klinkenberg, Sabine and Volkmann, Lutz

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:229b8b71-dbd2-4192-a971-d2ac8aafe0c9
uuid:229b8b71-dbd2-4192-a971-d2ac8aafe0c9

Subject:

matching, maximum matching, and close to regular graph

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

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.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

907-918

Source:

Czechoslovak Mathematical Journal | 2007 Volume:57 | Number:3

Harvested from:

CDK

Metadata only:

false