332 research outputs found
Semidefinite Programming and Ramsey Numbers
We use the theory of flag algebras to find new upper bounds for several small graph and hypergraph Ramsey numbers. In particular, we prove the exact values R(K−, K−, K−) = 28, R(K8, C5) = 29, R(K9, C6) = 41, R(Q3, Q3) = 13, R(K3,5, K1,6) = 17, R(C3, C5, C5) = 17, and R(K−, K−; 3) = 12, and in addition improve many additional upper bounds
Inverse Tur\'an numbers
For given graphs and , the Tur\'an number is defined to be
the maximum number of edges in an -free subgraph of . Foucaud,
Krivelevich and Perarnau and later independently Briggs and Cox introduced a
dual version of this problem wherein for a given number , one maximizes the
number of edges in a host graph for which .
Addressing a problem of Briggs and Cox, we determine the asymptotic value of
the inverse Tur\'an number of the paths of length and and provide an
improved lower bound for all paths of even length. Moreover, we obtain bounds
on the inverse Tur\'an number of even cycles giving improved bounds on the
leading coefficient in the case of . Finally, we give multiple conjectures
concerning the asymptotic value of the inverse Tur\'an number of and
, suggesting that in the latter problem the asymptotic behavior
depends heavily on the parity of .Comment: updated to include the suggestions of reviewer
Combinatorial theorems relative to a random set
We describe recent advances in the study of random analogues of combinatorial
theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201
Defensive alliances in graphs: a survey
A set of vertices of a graph is a defensive -alliance in if
every vertex of has at least more neighbors inside of than outside.
This is primarily an expository article surveying the principal known results
on defensive alliances in graph. Its seven sections are: Introduction,
Computational complexity and realizability, Defensive -alliance number,
Boundary defensive -alliances, Defensive alliances in Cartesian product
graphs, Partitioning a graph into defensive -alliances, and Defensive
-alliance free sets.Comment: 25 page
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
Local Hadwiger's Conjecture
We propose local versions of Hadwiger's Conjecture, where only balls of
radius around each vertex are required to be
-minor-free. We ask: if a graph is locally--minor-free, is it
-colourable? We show that the answer is yes when , even in the
stronger setting of list-colouring, and we complement this result with a
-round distributed colouring algorithm in the LOCAL model.
Further, we show that for large enough values of , we can list-colour
locally--minor-free graphs with colours, where is any value
such that all -minor-free graphs are -list-colourable. We again
complement this with a -round distributed algorithm.Comment: 24 pages; some minor typos have been fixe
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