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This paper is concerned with symmetric $1$-dependent colorings of the $d$-ray star graph $mathscr{S}^d$ for each $d ge 2$. We compute the critical point of the $1$-dependent hard-core processes on $mathscr{S}^d$, which gives a lower bound for the number of colors needed for a $1$-dependent coloring of $mathscr{S}^d$. We provide an explicit construction of a $1$-dependent $q$-coloring for any $q ge 5$ of the infinite subgraph $mathscr{S}^3_{(1,1,infty)}$, which is symmetric in the colors and whose restriction to any copy of $mathbb{Z}$ is some symmetric $1$-dependent $q$-coloring of $mathbb{Z}$. We also prove that there is no such coloring of $mathscr{S}^3_{(1,1,infty)}$ with $q = 4$ colors. A list of open problems are presented.
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In a recent paper by the same authors, we constructed a stationary 1-dependent 4-coloring of the integers that is invariant under permutations of the colors. This was the first stationary k-dependent q-coloring for any k and q. When the analogous construction is carried out for q>4 colors, the resulting process is not k-dependent for any k. We construct here a process that is symmetric in the colors and 1-dependent for every q>=4. The construction uses a recursion involving Chebyshev polynomials evaluated at $sqrt{q}/2$.
We study two weighted graph coloring problems, in which one assigns $q$ colors to the vertices of a graph such that adjacent vertices have different colors, with a vertex weighting $w$ that either disfavors or favors a given color. We exhibit a weighted chromatic polynomial $Ph(G,q,w)$ associated with this problem that generalizes the chromatic polynomial $P(G,q)$. General properties of this polynomial are proved, and illustrative calculations for various families of graphs are presented. We show that the weighted chromatic polynomial is able to distinguish between certain graphs that yield the same chromatic polynomial. We give a general structural formula for $Ph(G,q,w)$ for lattice strip graphs $G$ with periodic longitudinal boundary conditions. The zeros of $Ph(G,q,w)$ in the $q$ and $w$ planes and their accumulation sets in the limit of infinitely many vertices of $G$ are analyzed. Finally, some related weighted graph coloring problems are mentioned.
Very sparse random graphs are known to typically be singular (i.e., have singular adjacency matrix), due to the presence of low-degree dependencies such as isolated vertices and pairs of degree-1 vertices with the same neighbourhood. We prove that these kinds of dependencies are in some sense the only causes of singularity: for constants $kge 3$ and $lambda > 0$, an ErdH os--Renyi random graph $Gsimmathbb{G}(n,lambda/n)$ with $n$ vertices and edge probability $lambda/n$ typically has the property that its $k$-core (its largest subgraph with minimum degree at least $k$) is nonsingular. This resolves a conjecture of Vu from the 2014 International Congress of Mathematicians, and adds to a short list of known nonsingularity theorems for extremely sparse random matrices with density $O(1/n)$. A key aspect of our proof is a technique to extract high-degree vertices and use them to boost the rank, starting from approximate rank bounds obtainable from (non-quantitative) spectral convergence machinery due to Bordenave, Lelarge and Salez.
A wide array of random graph models have been postulated to understand properties of observed networks. Typically these models have a parameter $t$ and a critical time $t_c$ when a giant component emerges. It is conjectured that for a large class of models, the nature of this emergence is similar to that of the ErdH{o}s-Renyi random graph, in the sense that (a) the sizes of the maximal components in the critical regime scale like $n^{2/3}$, and (b) the structure of the maximal components at criticality (rescaled by $n^{-1/3}$) converges to random fractals. To date, (a) has been proven for a number of models using different techniques. This paper develops a general program for proving (b) that requires three ingredients: (i) in the critical scaling window, components merge approximately like the multiplicative coalescent, (ii) scaling exponents of susceptibility functions are the same as that of the ErdH{o}s-Renyi random graph, and (iii) macroscopic averaging of distances between vertices in the barely subcritical regime. We show that these apply to two fundamental random graph models: the configuration model and inhomogeneous random graphs with a finite ground space. For these models, we also obtain new results for component sizes at criticality and structural properties in the barely subcritical regime.
We study random walks on the giant component of the ErdH{o}s-Renyi random graph ${cal G}(n,p)$ where $p=lambda/n$ for $lambda>1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald, to have order $log^2 n$. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to $O(log n)$ and concentrates it (the cutoff phenomenon occurs): the typical mixing is at $( u {bf d})^{-1}log n pm (log n)^{1/2+o(1)}$, where $ u$ and ${bf d}$ are the speed of random walk and dimension of harmonic measure on a ${rm Poisson}(lambda)$-Galton-Watson tree. Analogous results are given for graphs with prescribed degree sequences, where cutoff is shown both for the simple and for the non-backtracking random walk.
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