## Taiwanese Journal of Mathematics

### MAXIMUM PACKINGS AND MINIMUM COVERINGS OF MULTIGRAPHS WITH PATHS AND STARS

#### Abstract

Let $F$, $G$, and $H$ be multigraphs. An $(F,G)$-decomposition of $H$ is an edge decomposition of $H$ into copies of $F$ and $G$ using at least one of each. For subgraphs $L$ and $R$ of $H$, an $(F,G)$-packing of $H$ with leave $L$ is an $(F,G)$-decomposition of $H-E(L)$, and an $(F,G)$-covering of $H$ with padding $R$ is an $(F,G)$-decomposition of $H+E(R)$. A maximum $(F,G)$-packing of $H$ is an $(F,G)$-packing of $H$ with a minimum leave. A minimum $(F,G)$-covering of $H$ is an $(F,G)$-covering of $H$ with a minimum padding.  Let $k$ be a positive integer. A $k$-path, denoted by $P_k$, is a path on $k$ vertices. A $k$-star, denoted by $S_k$, is a star with $k$ edges. In this paper, we obtain a maximum $(P_{k+1},S_{k})$-packing of $\lambda K_n$, which has a leave of size $\lt k$, and a minimum $(P_{k+1},S_{k})$-covering of $\lambda K_n$, which has a padding of size $\lt k$. A similar result for $\lambda K_{n,n}$ is also obtained. As corollaries, necessary and sufficient conditions for the existence of $(P_{k+1},S_k)$-decompositions of both $\lambda K_n$ and $\lambda K_{n,n}$ are given.

#### Article information

Source
Taiwanese J. Math., Volume 19, Number 5 (2015), 1341-1357.

Dates
First available in Project Euclid: 4 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499133712

Digital Object Identifier
doi:10.11650/tjm.19.2015.4456

Mathematical Reviews number (MathSciNet)
MR3412009

Zentralblatt MATH identifier
1357.05124

Keywords
packing covering path star

#### Citation

Lee, Hung-Chih; Chen, Zhen-Chun. MAXIMUM PACKINGS AND MINIMUM COVERINGS OF MULTIGRAPHS WITH PATHS AND STARS. Taiwanese J. Math. 19 (2015), no. 5, 1341--1357. doi:10.11650/tjm.19.2015.4456. https://projecteuclid.org/euclid.twjm/1499133712

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