Compound Node-Kayles on Paths

In his celebrated book "On Number and Games" (Academic Press, New-York, 1976), J.H. Conway introduced twelve versions of compound games. We analyze these twelve versions for the Node-Kayles game on paths. For usual disjunctive compound, Node-Kayles has been solved for a long time under normal play, while it is still unsolved under mis\`ere play. We thus focus on the ten remaining versions, leaving only one of them unsolved.Comment: Theoretical Computer Science (2009) to appea

On the Distinguishing Number of Cyclic Tournaments: Towards the Albertson-Collins Conjecture

A distinguishing $r$-labeling of a digraph $G$ is a mapping $\lambda$ from the set of verticesof $G$ to the set of labels $\{1,\dots,r\}$ such that no nontrivial automorphism of $G$ preserves all the labels.The distinguishing number $D(G)$ of $G$ is then the smallest $r$ for which $G$ admits a distinguishing $r$-labeling.From a result of Gluck (David Gluck, Trivial set-stabilizers in finite permutation groups,{\em Can. J. Math.} 35(1) (1983), 59--67),it follows that $D(T)=2$ for every cyclic tournament~$T$ of (odd) order $2q+1\ge 3$.Let $V(T)=\{0,\dots,2q\}$ for every such tournament.Albertson and Collins conjectured in 1999that the canonical 2-labeling $\lambda^*$ given by$\lambda^*(i)=1$ if and only if $i\le q$ is distinguishing.We prove that whenever one of the subtournaments of $T$ induced by vertices $\{0,\dots,q\}$or $\{q+1,\dots,2q\}$ is rigid, $T$ satisfies Albertson-Collins Conjecture.Using this property, we prove that several classes of cyclic tournaments satisfy Albertson-Collins Conjecture.Moreover, we also prove that every Paley tournament satisfies Albertson-Collins Conjecture

The bubble wall velocity in the minimal supersymmetric light stop scenario

We build on existing calculations of the wall velocity of the expanding bubbles of the broken symmetry phase in a first-order electroweak phase transition within the light stop scenario (LSS) of the MSSM. We carry out the analysis using the 2-loop thermal potential for values of the Higgs mass consistent with present experimental bounds. Our approach relies on describing the interaction between the bubble and the hot plasma by a single friction parameter, which we fix by matching to an existing 1-loop computation and extrapolate it to our regime of interest. For a sufficiently strong phase transition (in which washout of the newly-created baryon asymmetry is prevented) we obtain values of the wall velocity, v_w~0.05, far below the speed of sound in the medium, and not very much deviating from the previous 1-loop calculation. We also find that the phase transition is about 10% stronger than suggested by simply evaluating the thermal potential at the critical temperature.Comment: 17pages, 3 figure