Finding induced acyclic subgraphs in random digraphs.
Paths of specified length in random -partite graphs.
On sampling colorings of bipartite graphs.
Graphs of low chordality.
Induced Acyclic Tournaments in Random Digraphs: Sharp Concentration, Thresholds and Algorithms
Given a simple directed graph D = (V,A), let the size of the largest induced acyclic tournament be denoted by mat(D). Let D ∈ D(n, p) (with p = p(n)) be a random instance, obtained by randomly orienting each edge of a random graph drawn from Ϟ(n, 2p). We show that mat(D) is asymptotically almost surely (a.a.s.) one of only 2 possible values, namely either b*or b* + 1, where b* = ⌊2(logrn) + 0.5⌋ and r = p−1. It is also shown that if, asymptotically, 2(logrn) + 1 is not within a distance of w(n)/(ln...
Star Coloring of Subcubic Graphs
A star coloring of an undirected graph G is a coloring of the vertices of G such that (i) no two adjacent vertices receive the same color, and (ii) no path on 4 vertices is bi-colored. The star chromatic number of G, χs(G), is the minimum number of colors needed to star color G. In this paper, we show that if a graph G is either non-regular subcubic or cubic with girth at least 6, then χs(G) ≤ 6, and the bound can be realized in linear time.
On k-intersection edge colourings
We propose the following problem. For some k ≥ 1, a graph G is to be properly edge coloured such that any two adjacent vertices share at most k colours. We call this the k-intersection edge colouring. The minimum number of colours sufficient to guarantee such a colouring is the k-intersection chromatic index and is denoted χ’ₖ(G). Let fₖ be defined by . We show that fₖ(Δ) = Θ(Δ²/k). We also discuss some open problems.
Oriented colouring of some graph products
We obtain some improved upper and lower bounds on the oriented chromatic number for different classes of products of graphs.
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