arXiv subjects

Ross J. Kang

Publications and source records attributed to Ross J. Kang.

59 records · Page 4Linked to original sources

arXiv2012

Largest sparse subgraphs of random graphs

For the Erdős-Rényi random graph G(n,p), we give a precise asymptotic formula for the size of a largest vertex subset in G(n,p) that induces a subgraph with average degree at most t, provided that p = p(n) is not too small and t = t(n) is not too large. In the case of fixed t and p, we find that this value is asymptotically almost surely concentrated on at most two explicitly given points. This generalises a result on the independence number of random graphs. For both the upper and lower bounds, we rely on large deviations inequalities for the binomial distribution.

math.CO↗

arXiv2011

Every plane graph of maximum degree 8 has an edge-face 9-colouring

An edge-face colouring of a plane graph with edge set $E$ and face set $F$ is a colouring of the elements of $E \cup F$ such that adjacent or incident elements receive different colours. Borodin proved that every plane graph of maximum degree $Δ\ge10$ can be edge-face coloured with $Δ+1$ colours. Borodin's bound was recently extended to the case where $Δ=9$. In this paper, we extend it to the case $Δ=8$.

math.CO↗

arXiv2011

Rapid mixing of subset Glauber dynamics on graphs of bounded tree-width

Motivated by the `subgraphs world' view of the ferromagnetic Ising model, we develop a general approach to studying mixing times of Glauber dynamics based on subset expansion expressions for a class of graph polynomials. With a canonical paths argument, we demonstrate that the chains defined within this framework mix rapidly upon graphs of bounded tree-width. This extends known results on rapid mixing for the Tutte polynomial, the adjacency-rank ($R_2$-)polynomial and the interlace polynomial.

math.CO↗

arXiv2010

The t-stability number of a random graph

Given a graph G = (V,E), a vertex subset S is called t-stable (or t-dependent) if the subgraph G[S] induced on S has maximum degree at most t. The t-stability number of G is the maximum order of a t-stable set in G. We investigate the typical values that this parameter takes on a random graph on n vertices and edge probability equal to p. For any fixed 0 < p < 1 and fixed non-negative integer t, we show that, with probability tending to 1 as n grows, the t-stability number takes on at most two values which we identify as functions of t, p and n. The main tool we use is an asymptotic expression for the expected number of t-stable sets of order k. We derive this expression by performing a precise count of the number of graphs on k vertices that have maximum degree at most k. Using the above results, we also obtain asymptotic bounds on the t-improper chromatic number of a random graph (this is the generalisation of the chromatic number, where we partition of the vertex set of the graph into t-stable sets).

math.CO↗

arXiv2009

The t-improper chromatic number of random graphs

We consider the $t$-improper chromatic number of the Erd{\H o}s-R{é}nyi random graph $G(n,p)$. The t-improper chromatic number $χ^t(G)$ of $G$ is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most $t$. If $t = 0$, then this is the usual notion of proper colouring. When the edge probability $p$ is constant, we provide a detailed description of the asymptotic behaviour of $χ^t(G(n,p))$ over the range of choices for the growth of $t = t(n)$.

math.CO↗