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Graph Saturation in Multipartite Graphs
Let be a fixed graph and let be a family of graphs. A
subgraph of is -saturated if no member of
is a subgraph of , but for any edge in , some element of
is a subgraph of . We let and
denote the maximum and minimum size of an
-saturated subgraph of , respectively. If no element of
is a subgraph of , then .
In this paper, for and we determine
, where is the complete balanced -partite
graph with partite sets of size . We also give several families of
constructions of -saturated subgraphs of for . Our results
and constructions provide an informative contrast to recent results on the
edge-density version of from [A. Bondy, J. Shen, S.
Thomass\'e, and C. Thomassen, Density conditions for triangles in multipartite
graphs, Combinatorica 26 (2006), 121--131] and [F. Pfender, Complete subgraphs
in multipartite graphs, Combinatorica 32 (2012), no. 4, 483--495].Comment: 16 pages, 4 figure
Rainbow saturation and graph capacities
The -colored rainbow saturation number is the minimum size
of a -edge-colored graph on vertices that contains no rainbow copy of
, but the addition of any missing edge in any color creates such a rainbow
copy. Barrus, Ferrara, Vandenbussche and Wenger conjectured that for every and . In this short
note we prove the conjecture in a strong sense, asymptotically determining the
rainbow saturation number for triangles. Our lower bound is probabilistic in
spirit, the upper bound is based on the Shannon capacity of a certain family of
cliques.Comment: 5 pages, minor change
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