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On fan-saturated graphs

Jessica Fuller and Ronald J. Gould

Vol. 16 (2023), No. 4, 637–657
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 eE(G) would result in at least one copy of H as a subgraph. Let Ft 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 Ft-saturated graph for t 2. This set is called the saturation spectrum of Ft. For example, there exists an F2-saturated graph G on n 10 vertices and m edges if m = n + 2, or

2n 3 m n + 5 2 n 5 2 + 3n 5 2 + 4,

or m = p(n p) + 1, the size of the complete bipartite graph with one additional edge, or m = n24 n x2 + 1, x 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