Ross J. Kang

Induced matchings in subcubic planar graphs

We present a linear-time algorithm that, given a planar graph with m edges and maximum degree 3, finds an induced matching of size at least m/9. This is best possible.

Tight inequalities among set hitting times in Markov chains

Given an irreducible discrete time Markov chain on a finite state space, we consider the largest expected hitting time T(α) of a set of stationary measure at least α for α ∈ (0, 1). We obtain tight inequalities among the values of T(α) for different choices of α. One consequence is that T(α) ≤ T(1/2)/α for all α < 1/2. As a corollary we have that if the chain is lazy in a certain sense as well as reversible, then T(1/2) is equivalent to the chain’s mixing time, answering a question of Peres. We furthermore demonstrate that the inequalities we establish give an almost everywhere pointwise limiting characterisation of possible hitting time functions T(α) over the domain α ∈ (0, 1/2].

On r -dynamic coloring of grids

An r -dynamic k -coloring of a graph G is a proper k -coloring of G such that every vertex in V ( G ) has neighbors in at least min { d ( v ) , r } different color classes. The r -dynamic chromatic number of a graph G , written ? r ( G ) , is the least k such that G has such a coloring. Proving a conjecture of Jahanbekam, Kim, O, and West, we show that the m -by- n grid has no 3-dynamic 4-coloring when m n ? 2 mod 4 (for m , n ? 3 ). This completes the determination of the r -dynamic chromatic number of the m -by- n grid for all r , m , n .

Acyclic improper colourings of graphs with bounded maximum degree

For graphs of bounded maximum degree, we consider acyclic t-improper colourings, that is, colourings in which each bipartite subgraph consisting of the edges between two colour classes is acyclic, and each colour class induces a graph with maximum degree at most t. We consider the supremum, over all graphs of maximum degree at most d, of the acyclic t-improper chromatic number and provide t-improper analogues of results by Alon, McDiarmid and Reed [N. Alon, C.J.H. McDiarmid, B. Reed, Acyclic coloring of graphs, Random Structures Algorithms 2 (3) (1991) 277-288] and Fertin, Raspaud and Reed [G. Fertin, A. Raspaud, B. Reed, Star coloring of graphs, J. Graph Theory 47 (3) (2004) 163-182].

The Distance-t Chromatic Index of Graphs

We consider two graph colouring problems in which edges at distance at most t are given distinct colours, for some fixed positive integer t . We obtain two upper bounds for the distance- t chromatic index, the least number of colours necessary for such a colouring. One is a bound of (2-e)Δ t for graphs of maximum degree at most Δ, where e is some absolute positive constant independent of t . The other is a bound of O (Δ t /log Δ) (as Δ → ∞) for graphs of maximum degree at most Δ and girth at least 2 t +1. The first bound is an analogue of Molloy and Reeds bound on the strong chromatic index. The second bound is tight up to a constant multiplicative factor, as certified by a class of graphs of girth at least g , for every fixed g ≥ 3, of arbitrarily large maximum degree Δ, with distance- t chromatic index at least Ω(Δ t /log Δ).

Improper Colourings of Unit Disk Graphs

We investigate the following problem proposed by Alcatel. A satellite sends information to receivers on earth, each of which is listening on a chosen frequency. Technically, it is impossible for the satellite to precisely focus its signal onto a receiver. Part of the signal will be spread in an area around its destination and this creates noise for nearby receivers on the same frequency. However, a receiver is able to distinguish its signal if the sum of the noise does not become too large, i.e. does not exceed a certain threshold T . The problem

Improper choosability and Property B

A fundamental connection between list vertex colorings of graphs and Property B (also known as hypergraph 2-colorability)was already known to Erd˝os, Rubin, and Taylor ((1980), 125–157). In this article, we draw similar connections for improper list colorings. This extends results of Kostochka (Electron J Combin 9 (2002)), Alon (Combin Probab Comput 1 (1992), 107–114; (1993), 1–33; Random Structures Algorithms 16 (2000), 364–368), and Kr´ al’ and Sgall (J Graph Theory 49 (2005), 177–186) for, respectively, multipartite graphs, graphs of large minimum degree, and list assignments with bounded list union.

A Precise Threshold For Quasi-Ramsey Numbers

We consider the variation of Ramsey numbers introduced by Erdos and Pach [J. Graph Theory, 7 (1983), pp. 137--147], where instead of seeking complete or independent sets we only seek a

Sphere and dot product representations of graphs

t

Every Plane Graph of Maximum Degree 8 has an Edge-Face 9-Coloring

-homogeneous set, a vertex subset that induces a subgraph of minimum degree at least