On fan-saturated graphs

Fuller, Jessica; Gould, Ronald J.

doi:10.2140/involve.2023.16.637

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On fan-saturated graphs

Jessica Fuller and Ronald J. Gould

Vol. 16 (2023), No. 4, 637–657

DOI: 10.2140/involve.2023.16.637

Abstract

Given a graph $H$ , we say that a graph $G$ is $H$ -saturated if it does not contain $H$ as a subgraph, but the addition of any edge $e \notin E (G)$ would result in at least one copy of $H$ as a subgraph. Let $F_{t}$ be the graph consisting of $t$ edge-disjoint triangles that intersect at a single vertex $v$ . We investigate the set of all $m$ such that there exists an $n$ -vertex, $m$ -edge $F_{t}$ -saturated graph for $t \geq 2$ . This set is called the saturation spectrum of $F_{t}$ . For example, there exists an $F_{2}$ -saturated graph $G$ on $n \geq 10$ vertices and $m$ edges if $m = n + 2$ , or

2 n - 3 \leq m \leq ⌈ \frac{n + 5}{2} ⌉ ⌊ \frac{n - 5}{2} ⌋ + 3 ⌊ \frac{n - 5}{2} ⌋ + 4,

or $m = p (n - p) + 1$ , the size of the complete bipartite graph with one additional edge, or $m = n^{2} ∕ 4 - n - x^{2} + 1$ , $x \geq 1$ .

Keywords

fan, saturation, saturation spectrum

Mathematical Subject Classification

Primary: 05C35

Milestones

Received: 17 May 2022

Revised: 3 October 2022

Accepted: 6 October 2022

Published: 31 October 2023

Communicated by Ann N. Trenk

Authors

Jessica Fuller
Department of Mathematics University of Connecticut Stamford, CT United States

Ronald J. Gould
Department of Mathematics and Computer Science Emory University Atlanta, GA United States

Vol. 16, No. 4, 2023