1,221 research outputs found
On the editing distance of graphs
An edge-operation on a graph is defined to be either the deletion of an
existing edge or the addition of a nonexisting edge. Given a family of graphs
, the editing distance from to is the smallest
number of edge-operations needed to modify into a graph from .
In this paper, we fix a graph and consider , the set of
all graphs on vertices that have no induced copy of . We provide bounds
for the maximum over all -vertex graphs of the editing distance from
to , using an invariant we call the {\it binary chromatic
number} of the graph . We give asymptotically tight bounds for that distance
when is self-complementary and exact results for several small graphs
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
Packing spanning graphs from separable families
Let be a separable family of graphs. Then for all positive
constants and and for every sufficiently large integer ,
every sequence of graphs of order and maximum
degree at most such that packs into . This improves results of
B\"ottcher, Hladk\'y, Piguet, and Taraz when is the class of trees
and of Messuti, R\"odl, and Schacht in the case of a general separable family.
The result also implies approximate versions of the Oberwolfach problem and of
the Tree Packing Conjecture of Gy\'arf\'as (1976) for the case that all trees
have maximum degree at most . The proof uses the local resilience of
random graphs and a special multi-stage packing procedure
Ramsey properties of randomly perturbed graphs: cliques and cycles
Given graphs , a graph is -Ramsey if for every
colouring of the edges of with red and blue, there is a red copy of
or a blue copy of . In this paper we investigate Ramsey questions in the
setting of randomly perturbed graphs: this is a random graph model introduced
by Bohman, Frieze and Martin in which one starts with a dense graph and then
adds a given number of random edges to it. The study of Ramsey properties of
randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali in
2006; they determined how many random edges must be added to a dense graph to
ensure the resulting graph is with high probability -Ramsey (for
). They also raised the question of generalising this result to pairs
of graphs other than . We make significant progress on this
question, giving a precise solution in the case when and
where . Although we again show that one requires polynomially fewer
edges than in the purely random graph, our result shows that the problem in
this case is quite different to the -Ramsey question. Moreover, we
give bounds for the corresponding -Ramsey question; together with a
construction of Powierski this resolves the -Ramsey problem.
We also give a precise solution to the analogous question in the case when
both and are cycles. Additionally we consider the
corresponding multicolour problem. Our final result gives another
generalisation of the Krivelevich, Sudakov and Tetali result. Specifically, we
determine how many random edges must be added to a dense graph to ensure the
resulting graph is with high probability -Ramsey (for odd
and ).Comment: 24 pages + 12-page appendix; v2: cited independent work of Emil
Powierski, stated results for cliques in graphs of low positive density
separately (Theorem 1.6) for clarity; v3: author accepted manuscript, to
appear in CP
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