Modern Graph Theory

pdf/correspondence-coloring-and-its-application-to-list-coloring-42n1se9k2r.pdf

Correspondence coloring and its application to list-coloring planar graphs without cycles of lengths 4 to 8

https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.081/Henning/matroidaxioms.pdf

Axioms for infinite matroids

We prove an optimal mixing time bound for the single-site update Markov chain known as the Glauber dynamics or Gibbs sampling in a variety of settings. Our work presents an improved version of the spectral independence approach of Anari et al. (2020) and shows O(nlogn) mixing time on any n-vertex graph of bounded degree when the maximum eigenvalue of an associated influence matrix is bounded. As an application of our results, for the hard-core model on independent sets weighted by a fugacity λ, we establish O(nlogn) mixing time for the Glauber dynamics on any n-vertex graph of constant maximum degree Δ when λc(Δ) where λc(Δ) is the critical point for the uniqueness/non-uniqueness phase transition on the Δ-regular tree. More generally, for any antiferromagnetic 2-spin system we prove O(nlogn) mixing time of the Glauber dynamics on any bounded degree graph in the corresponding tree uniqueness region. Our results apply more broadly; for example, we also obtain O(nlogn) mixing for q-colorings of triangle-free graphs of maximum degree Δ when the number of colors satisfies q > α Δ where α ≈ 1.763, and O(mlogn) mixing for generating random matchings of any graph with bounded degree and m edges. Our approach is based on two steps. First, we show that the approximate tensorization of entropy (i.e., factorizing entropy into single vertices), which is a key step for establishing the modified log-Sobolev inequality in many previous works, can be deduced from entropy factorization into blocks of fixed linear size. Second, we adapt the local-to-global scheme of Alev and Lau (2020) to establish such block factorization of entropy in a more general setting of pure weighted simplicial complexes satisfying local spectral expansion; this also substantially generalizes the result of Cryan et al. (2019).

Optimal mixing of Glauber dynamics: entropy factorization via high-dimensional expansion

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 mixing for $k\ge\Delta+2$. In FOCS 1999, Vigoda showed that the flip dynamics (and therefore also Glauber dynamics) is rapidly mixing for any $k>\frac{11}{6}\Delta$. It turns out that there is a natural barrier at $\frac{11}{6}$, below which there is no one-step coupling that is contractive with respect to the Hamming metric, even for the flip dynamics. 
We use linear programming and duality arguments to fully characterize the obstructions to going beyond $\frac{11}{6}$. These extremal configurations turn out to be quite brittle, and in this paper we use this to give two proofs that the Glauber dynamics is rapidly mixing for any $k\ge\left(\frac{11}{6} - \epsilon_0\right)\Delta$ for some absolute constant $\epsilon_0>0$. This is the first improvement to Vigoda's result that holds for general graphs. Our first approach analyzes a variable-length coupling in which these configurations break apart with high probability before the coupling terminates, and our other approach analyzes a one-step path coupling with a new metric that counts the extremal configurations. Additionally, our results extend to list coloring, a widely studied generalization of coloring, where the previously best known results required $k > 2 \Delta$.

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 Δ is rapidly mixing for k ≥ Δ + 2. In FOCS 1999, Vigoda [43] showed that the flip dynamics (and therefore also Glauber dynamics) is rapidly mixing for any [MATH HERE]. It turns out that there is a natural barrier at [MATH HERE], below which there is no one-step coupling that is contractive with respect to the Hamming metric, even for the flip dynamics. We use linear programming and duality arguments to fully characterize the obstructions to going beyond [MATH HERE]. These extremal configurations turn out to be quite brittle, and in this paper we use this to give two proofs that the Glauber dynamics is rapidly mixing for any [MATH HERE] for some absolute constant ϵe0 > 0. This is the first improvement to Vigoda's result that holds for general graphs. Our first approach analyzes a variable-length coupling in which these configurations break apart with high probability before the coupling terminates, and our other approach analyzes a one-step path coupling with a new metric that counts the extremal configurations. Additionally, our results extend to list coloring, a widely studied generalization of coloring, where the previously best known results required k > 2Δ.

/pdf/improved-bounds-for-randomly-sampling-colorings-via-linear-46g9cx89cc.pdf

Improved bounds for randomly sampling colorings via linear programming

A graph G is 5/2-critical if G has no circular 5/2-coloring (or equivalently, homomorphism to C5), but every proper subgraph of G has one. We prove that every 5/2-critical graph on n ≥ 4 vertices has at least 

$$\frac{{5n - 2}}{4}$$

edges, and list all 5/2-critical graphs achieving this bound. This implies that every planar or projective-planar graph of girth at least 10 is 5/2-colorable.

/pdf/density-of-5-2-critical-graphs-3y2x9z8sy4.pdf

Density of 5/2-critical graphs

Let $G$ be a graph embedded in a fixed surface $\Sigma$ of genus $g$ and let $L=(L(v):v\in V(G))$ be a collection of lists such that either each list has size at least five, or each list has size at least four and $G$ is triangle-free, or each list has size at least three and $G$ has no cycle of length four or less. An $L$-coloring of $G$ is a mapping $\phi$ with domain $V(G)$ such that $\phi(v)\in L(v)$ for every $v\in V(G)$ and $\phi(v)\ne\phi(u)$ for every pair of adjacent vertices $u,v\in V(G)$. We prove 
* if every non-null-homotopic cycle in $G$ has length $\Omega(\log g)$, then $G$ has an $L$-coloring, 
* if $G$ does not have an $L$-coloring, but every proper subgraph does ("$L$-critical graph"), then $|V(G)|=O(g)$, 
* if every non-null-homotopic cycle in $G$ has length $\Omega(g)$, and a set $X\subseteq V(G)$ of vertices that are pairwise at distance $\Omega(1)$ is precolored from the corresponding lists, then the precoloring extends to an $L$-coloring of $G$, 
* if every non-null-homotopic cycle in $G$ has length $\Omega(g)$, and the graph $G$ is allowed to have crossings, but every two crossings are at distance $\Omega(1)$, then $G$ has an $L$-coloring, and 
* if $G$ has at least one $L$-coloring, then it has at least $2^{\Omega(|V(G)|)}$ distinct $L$-colorings. 
We show that the above assertions are consequences of certain isoperimetric inequalities satisfied by $L$-critical graphs, and we study the structure of families of embedded graphs that satisfy those inequalities. It follows that the above assertions hold for other coloring problems, as long as the corresponding critical graphs satisfy the same inequalities.

Hyperbolic families and coloring graphs on surfaces

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