Saturation numbers for families of graph subdivisions

Abstract

For a family scriptF of graphs, a graph G is scriptF‐saturated if G contains no member of scriptF as a subgraph, but for any edge uv in G¯, G+uv contains some member of scriptF as a subgraph. The minimum number of edges in an scriptF‐saturated graph of order n is denoted sat(n,F). A subdivision of a graph H, or an H‐subdivision, is a graph G obtained from H by replacing the edges of H with internally disjoint paths of arbitrary length. We let S(H) denote the family of H‐subdivisions, including H itself. In this paper, we study sat(n,S(H)) when H is one of Ct or Kt, obtaining several exact results and bounds. In particular, we determine sat(n,S(Ct)) exactly for 3≤t≤5 and show for n sufficiently large that there exists a constant ct such that 54n≤sat(n,scriptS(Ct))≤54+cttn. For t≥36 we show that ct=8 will suffice, and that this can be improved slightly depending on the value of t(mod8). We also give an upper bound on sat(n,S(Kt)) for all t and show that sat(n,scriptS(K5))=⌈3n+42⌉. This provides an interesting contrast to a 1937 result of Wagner (Math Ann, 114 (1937), 570–590), who showed that edge‐maximal graphs without a K5‐minor have at least 11n6 edges.

Saturation for the $3$-uniform loose $3$-cycle

“…This highlights a difficulty in studying the saturation function in general and for cycles -one usually does not expect to prove a nice stability result when there are multiple different extremal examples. Saturation numbers for the family of all cycles of length above a certain value have also been studied, with exact results determined for 3 ≤ ≤ 6 [15,22]. In addition to these results involving cycles of short length, many results on the saturation numbers of Hamiltonian cycles have been studied, with numerous results leading up to proving that sat 2 (n, C n ) = 3n 2 (upper bound given first in [4], while the lower bound can be found in [21]).…”

“…So, ∈ N (u), g 2 has the form {v, x, w} for some vertex w. If w = v 1 , then e 1 , e 2 and g 2 form a triangle in G. Hence w = v 1 . Since each of e 2 and g 2 contains only one vertex in M , (15)…”

Induced Subgraph Saturated Graphs

“…with addition modulo k, the external cycle ( Figure 6). Figure 6: The graph G [5], which is induced C 7 -saturated…”

“…Since then, graph saturation has been studied extensively, having been generalized to many other families of graphs, oriented graphs [7], topological minors [5], and numerous other properties. A comprehensive collection of results in graph saturation is available in [4].…”

Saturation Numbers for Minors

“…Now let S(H) be the family of subdivisions of H; it follows that sat(n, M(C r )) = sat(n, S(C r )). It is known that there exists a constant c so that 5 4 n ≤ sat(n, S(C r )) ≤ ( 5 4 + c r 2 )n + O(1) [1]. Finally, sat(n, M(K 3 )) = n − 1, sat(n, M(K 4 )) = 2n − 3, and sat(n, M(K 5 )) = 11 6 n + O(1); these values are easily obtained via well known characterizations of graphs which don't contain K 3 , K 4 , or K 5 as minors.…”