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Improved Bounds for Randomly Sampling Colorings via Linear Programming
A well-known conjecture in computer science and statistical physics is that Glauber dynamics on the set of $k$-colorings of a graph $G$ on $n$ vertices with maximum degree $\Delta$ is rapidly mixingExpand
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Progress towards Nash-Williams' Conjecture on Triangle Decompositions
Partitioning the edges of a graph into edge disjoint triangles forms a triangle decomposition of the graph. A famous conjecture by Nash-Williams from 1970 asserts that any sufficiently large,Expand
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Intersecting families of discrete structures are typically trivial
The study of intersecting structures is central to extremal combinatorics. A family of permutations F ? S n is t-intersecting if any two permutations in F agree on some t indices, and is trivial ifExpand
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Locating a robber on a graph via distance queries
A cop wants to locate a robber hiding among the vertices of a graph. A round of the game consists of the robber moving to a neighbor of its current vertex (or not moving) and then the cop scanningExpand
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Rapid mixing of Glauber dynamics for colorings below Vigoda's 11/6 threshold
A well-known conjecture in computer science and statistical physics is that Glauber dynamics on the set of $k$-colorings of a graph $G$ on $n$ vertices with maximum degree $\Delta$ is rapidly mixingExpand
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On the List Coloring Version of Reed's Conjecture
Abstract The chromatic number of a graph is bounded below by its clique number and from above by its maximum degree plus one. In 1998, Reed conjectured that the chromatic number is at most halfway inExpand
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The typical structure of intersecting families of discrete structures
The study of intersecting structures is central to extremal combinatorics. A family of permutations F ⊂ Sn is t-intersecting if any two permutations in F agree on some t indices, and is trivial ifExpand
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Rainbow copies of C4 in edge-colored hypercubes
For positive integers k and d such that 4 ź k < d and k ź 5 , we determine the maximum number of rainbow colored copies of C 4 in a k -edge-coloring of the d -dimensional hypercube Q d .Expand
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On a Conjecture of Thomassen
In 1989, Thomassen asked whether there is an integer-valued function $f(k)$ such that every $f(k)$-connected graph admits a spanning, bipartite $k$-connected subgraph. In this paper we take a first,Expand
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A combinatorial method for connecting BHV spaces representing different numbers of taxa
The phylogenetic tree space introduced by Billera, Holmes, and Vogtmann (BHV tree space) is a CAT(0) continuous space that represents trees with edge weights with an intrinsic geodesic distanceExpand
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