3,909 research outputs found
On the minimum degree of minimal Ramsey graphs for multiple colours
A graph G is r-Ramsey for a graph H, denoted by G\rightarrow (H)_r, if every
r-colouring of the edges of G contains a monochromatic copy of H. The graph G
is called r-Ramsey-minimal for H if it is r-Ramsey for H but no proper subgraph
of G possesses this property. Let s_r(H) denote the smallest minimum degree of
G over all graphs G that are r-Ramsey-minimal for H. The study of the parameter
s_2 was initiated by Burr, Erd\H{o}s, and Lov\'{a}sz in 1976 when they showed
that for the clique s_2(K_k)=(k-1)^2. In this paper, we study the dependency of
s_r(K_k) on r and show that, under the condition that k is constant, s_r(K_k) =
r^2 polylog r. We also give an upper bound on s_r(K_k) which is polynomial in
both r and k, and we determine s_r(K_3) up to a factor of log r
Decomposition of bounded degree graphs into -free subgraphs
We prove that every graph with maximum degree admits a partition of
its edges into parts (as ) none of which
contains as a subgraph. This bound is sharp up to a constant factor. Our
proof uses an iterated random colouring procedure.Comment: 8 pages; to appear in European Journal of Combinatoric
Syntactic Separation of Subset Satisfiability Problems
Variants of the Exponential Time Hypothesis (ETH) have been used to derive lower bounds on the time complexity for certain problems, so that the hardness results match long-standing algorithmic results. In this paper, we consider a syntactically defined class of problems, and give conditions for when problems in this class require strongly exponential time to approximate to within a factor of (1-epsilon) for some constant epsilon > 0, assuming the Gap Exponential Time Hypothesis (Gap-ETH), versus when they admit a PTAS. Our class includes a rich set of problems from additive combinatorics, computational geometry, and graph theory. Our hardness results also match the best known algorithmic results for these problems
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