3-Colourability of Dually Chordal Graphs in Linear Time

A graph G is dually chordal if there is a spanning tree T of G such that any maximal clique of G induces a subtree in T. This paper investigates the Colourability problem on dually chordal graphs. It will show that it is NP-complete in case of four colours and solvable in linear time with a simple algorithm in case of three colours. In addition, it will be shown that a dually chordal graph is 3-colourable if and only if it is perfect and has no clique of size four

On Hamilton Decompositions of Line Graphs of Non-Hamiltonian Graphs and Graphs without Separating Transitions

In contrast with Kotzig's result that the line graph of a $3$-regular graph $X$ is Hamilton decomposable if and only if $X$ is Hamiltonian, we show that for each integer $k\geq 4$ there exists a simple non-Hamiltonian $k$-regular graph whose line graph has a Hamilton decomposition. We also answer a question of Jackson by showing that for each integer $k\geq 3$ there exists a simple connected $k$-regular graph with no separating transitions whose line graph has no Hamilton decomposition

The first order convergence law fails for random perfect graphs

We consider first order expressible properties of random perfect graphs. That is, we pick a graph $G_n$ uniformly at random from all (labelled) perfect graphs on $n$ vertices and consider the probability that it satisfies some graph property that can be expressed in the first order language of graphs. We show that there exists such a first order expressible property for which the probability that $G_n$ satisfies it does not converge as $n\to\infty$.Comment: 11 pages. Minor corrections since last versio

Revolutionaries and spies on random graphs

Pursuit-evasion games, such as the game of Revolutionaries and Spies, are a simplified model for network security. In the game we consider in this paper, a team of $r$ revolutionaries tries to hold an unguarded meeting consisting of $m$ revolutionaries. A team of $s$ spies wants to prevent this forever. For given $r$ and $m$, the minimum number of spies required to win on a graph $G$ is the spy number $\sigma(G,r,m)$. We present asymptotic results for the game played on random graphs $G(n,p)$ for a large range of $p = p(n), r=r(n)$, and $m=m(n)$. The behaviour of the spy number is analyzed completely for dense graphs (that is, graphs with average degree at least n^{1/2+\eps} for some \eps > 0). For sparser graphs, some bounds are provided