336 research outputs found
Color-blind index in graphs of very low degree
Let be an edge-coloring of a graph , not necessarily
proper. For each vertex , let , where is
the number of edges incident to with color . Reorder for
every in in nonincreasing order to obtain , the color-blind
partition of . When induces a proper vertex coloring, that is,
for every edge in , we say that is color-blind
distinguishing. The minimum for which there exists a color-blind
distinguishing edge coloring is the color-blind index of ,
denoted . We demonstrate that determining the
color-blind index is more subtle than previously thought. In particular,
determining if is NP-complete. We also connect
the color-blind index of a regular bipartite graph to 2-colorable regular
hypergraphs and characterize when is finite for a class
of 3-regular graphs.Comment: 10 pages, 3 figures, and a 4 page appendi
The Erd\H{o}s-Rothschild problem on edge-colourings with forbidden monochromatic cliques
Let be a sequence of natural numbers. For a
graph , let denote the number of colourings of the edges
of with colours such that, for every , the
edges of colour contain no clique of order . Write
to denote the maximum of over all graphs on vertices.
This problem was first considered by Erd\H{o}s and Rothschild in 1974, but it
has been solved only for a very small number of non-trivial cases.
We prove that, for every and , there is a complete
multipartite graph on vertices with . Also, for every we construct a finite
optimisation problem whose maximum is equal to the limit of as tends to infinity. Our final result is a
stability theorem for complete multipartite graphs , describing the
asymptotic structure of such with in terms of solutions to the optimisation problem.Comment: 16 pages, to appear in Math. Proc. Cambridge Phil. So
Extending Edge-Colorings of Complete Uniform Hypergraphs
A hypergraph is an ordered pair where is a set of vertices of and is a collection of edge multisets of . If the size of every edge in the hypergraph is equal, then we call it a uniform hypergraph. A \textit{complete -uniform hypergraph}, written , is a uniform hypergraph with edge sizes equal to and has vertices where the edges set is the collection of all -elements subset of its vertex set (so the total number of the edges is ). A hypergraph is called \textit{regular} if the degree of all vertices is the same. An -factorization of a hypergraph is a coloring of the edges of a hypergraph such that the number of times each element appears in each color class is exactly . A partial -factorization is a coloring in which the degree of each vertex in each color class is at most .
The main problem under consideration in this thesis is motivated by Baranyai\u27s famous theorem and Cameron\u27s question from 1976. Given a partial -factorization of , we are interested in finding the necessary and sufficient conditions under which we can extend this partial -factorization to an -factorization of . The case of this problem was partially solved by Bahmanian and Rodger in 2012, and the cases were partially solved by Bahmanian in 2018. Recently, Bahmanian and Johnsen showed that as long as , the obvious necessary conditions are also sufficient. In this thesis, we improve this bound for all . Our proof is computer-assisted
A reverse Sidorenko inequality
Let be a graph allowing loops as well as vertex and edge weights. We
prove that, for every triangle-free graph without isolated vertices, the
weighted number of graph homomorphisms satisfies the inequality
where denotes the degree of vertex in . In particular, one has for every -regular
triangle-free . The triangle-free hypothesis on is best possible. More
generally, we prove a graphical Brascamp-Lieb type inequality, where every edge
of is assigned some two-variable function. These inequalities imply tight
upper bounds on the partition function of various statistical models such as
the Ising and Potts models, which includes independent sets and graph
colorings.
For graph colorings, corresponding to , we show that the
triangle-free hypothesis on may be dropped; this is also valid if some of
the vertices of are looped. A corollary is that among -regular graphs,
maximizes the quantity for every and ,
where counts proper -colorings of .
Finally, we show that if the edge-weight matrix of is positive
semidefinite, then This implies that among -regular graphs,
maximizes . For 2-spin Ising models, our results give a
complete characterization of extremal graphs: complete bipartite graphs
maximize the partition function of 2-spin antiferromagnetic models and cliques
maximize the partition function of ferromagnetic models.
These results settle a number of conjectures by Galvin-Tetali, Galvin, and
Cohen-Csikv\'ari-Perkins-Tetali, and provide an alternate proof to a conjecture
by Kahn.Comment: 30 page
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