Partite saturation number of cycles

TL;DR

This paper investigates the minimum size of cycle-saturated subgraphs within complete multipartite graphs, providing asymptotic bounds and exact values for various cycle lengths and partite configurations.

Contribution

It offers the first asymptotically tight bounds and exact values for the partite saturation number of cycles in complete multipartite graphs.

Findings

Derived asymptotically tight bounds for $sat(K_k^n,C_ ext{)}$ for all  extgreater 4, except (4,4)

Determined exact values of $sat(K_k^n,C_ ext{)}$ for specific cases such as $k>=4$ and $5 extgreater  > k extgreater 3$

Provided solutions for the case $(,k)=(6,2)$

Abstract

A graph $H$ is said to be $F$-saturated relative to $G$, if $H$ does not contain any copy of $F$, but the addition of any edge $e$ in $E(G)\backslash E(H)$ would create a copy of $F$. The minimum size of an $F$-saturated graph relative to $G$ is denoted by $sat(G,F)$. Let $K_k^n$ be the complete $k$-partite graph containing $n$ vertices in each part and $C_\ell$ be the cycle of length $\ell$. In this paper we give an asymptotically tight bound of $sat(K_k^n,C_\ell)$ for all $ \ell \geq 4, k \geq 2$ except $(\ell,k)=(4,4)$. Moreover, we determined the exact value of $sat(K_k^n,C_\ell)$ for $ k>\ell=4 $ and $5 \geq \ell>k \geq 3$ and $(\ell,k)=(6,2)$.