Minimum saturated graphs for unions of cliques
Abstract
Let $H$ be a fixed graph. A graph $G$ is called {\it $H$saturated} if $H$ is not a subgraph of $G$ but the addition of any missing edge to $G$ results in an $H$subgraph. The {\it saturation number} of $H$, denoted $sat(n,H)$, is the minimum number of edges over all $H$saturated graphs of order $n$, and $Sat(n,H)$ denote the family of $H$saturated graphs with $sat(n,H)$ edges and $n$ vertices. In this paper, we resolve a conjecture of Chen and Yuan in[Discrete Math. 347(2024)113868] by determining $Sat(n,K_p\cup (t1)K_q)$ for every $2\le p\le q$ and $t\ge 2$.
 Publication:

arXiv eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.12204
 arXiv:
 arXiv:2404.12204
 Bibcode:
 2024arXiv240412204Z
 Keywords:

 Mathematics  Combinatorics;
 05C35
 EPrint:
 7 pages, 2 figures