Uniquely cycle-saturated graphs

TL;DR

This paper characterizes uniquely cycle-saturated graphs, showing that uniquely $C_5$-saturated graphs are friendship graphs, and proving finiteness or non-existence for other cycle lengths.

Contribution

It provides a complete characterization for uniquely $C_5$-saturated graphs and establishes finiteness or non-existence results for other cycle lengths, advancing understanding of cycle-saturated graph structures.

Findings

Uniquely $C_5$-saturated graphs are exactly friendship graphs.

No uniquely $C_6$- or $C_7$-saturated graphs exist.

Finitely many uniquely $C_t$-saturated graphs for $t extgreater6$, with a conjecture of none existing.

Abstract

Given a graph $F$, a graph $G$ is {\it uniquely $F$-saturated} if $F$ is not a subgraph of $G$ and adding any edge of the complement to $G$ completes exactly one copy of $F$. In this paper we study uniquely $C_t$-saturated graphs. We prove the following: (1) a graph is uniquely $C_5$-saturated if and only if it is a friendship graph. (2) There are no uniquely $C_6$-saturated graphs or uniquely $C_7$-saturated graphs. (3) For $t\ge6$, there are only finitely many uniquely $C_t$-saturated graphs (we conjecture that in fact there are none).