360 research outputs found
Almost spanning subgraphs of random graphs after adversarial edge removal
Let Delta>1 be a fixed integer. We show that the random graph G(n,p) with
p>>(log n/n)^{1/Delta} is robust with respect to the containment of almost
spanning bipartite graphs H with maximum degree Delta and sublinear bandwidth
in the following sense: asymptotically almost surely, if an adversary deletes
arbitrary edges in G(n,p) such that each vertex loses less than half of its
neighbours, then the resulting graph still contains a copy of all such H.Comment: 46 pages, 6 figure
Rainbow independent sets on dense graph classes
Given a family of independent sets in a graph, a rainbow
independent set is an independent set such that there is an injection
where for each , is contained in
. Aharoni, Briggs, J. Kim, and M. Kim [Rainbow independent sets in
certain classes of graphs. arXiv:1909.13143] determined for various graph
classes whether satisfies a property that for every
, there exists such that every family of
independent sets of size in a graph in contains a rainbow
independent set of size . In this paper, we add two dense graph classes
satisfying this property, namely, the class of graphs of bounded neighborhood
diversity and the class of -powers of graphs in a bounded expansion class
Note on Rainbow Connection in Oriented Graphs with Diameter 2
In this note, we provide a sharp upper bound on the rainbow connection number of tournaments of diameter . For a tournament of diameter , we show . Furthermore, we provide a general upper bound on the rainbow -connection number of tournaments as a simple example of the probabilistic method. Finally, we show that an edge-colored tournament of diameter has rainbow -connection number at most approximately
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