Extending partial colorings of graphs

AbstractRecently, several authors have considered the problem of extending a partial coloring of a graph to a complete coloring. We show how similar results can be extracted from old proofs on recursive colorings of highly recursive graphs

Edge coloring multigraphs without small dense subsets

© 2015. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/One consequence of a long-standing conjecture of Goldberg and Seymour about the chromatic index of multigraphs would be the following statement. Suppose $G$ is a multigraph with maximum degree $\Delta$, such that no vertex subset $S$ of odd size at most $\Delta$ induces more than $(\Delta+1)(|S|-1)/2$ edges. Then $G$ has an edge coloring with $\Delta+1$ colors. Here we prove a weakened version of this statement.Natural Sciences and Engineering Research Counci

Critical temperature of the superfluid transition in bose liquids

A phenomenological criterion for the superfluid transition is proposed, which is similar to the Lindemann criterion for the crystal melting. Then we derive a new formula for the critical temperature, relating $T_{\lambda}$ to the mean kinetic energy per particle above the transition. The suppression of the critical temperature in a sufficiently dense liquid is described as a result of the quantum decoherence phenomenon. The theory can account for the observed dependence of $T_{\lambda}$ on density in liquid helium and results in an estimate $T_{\lambda} \sim 1.1$ K for molecular hydrogen.Comment: 4 pages, 1 fi

Weak acyclic coloring and asymmetric coloring games

AbstractWe introduce the notion of weak acyclic coloring of a graph. This is a relaxation of the usual notion of acyclic coloring which is often sufficient for applications. We then use this concept to analyze the (a,b)-coloring game. This game is played on a finite graph G, using a set of colors X, by two players Alice and Bob with Alice playing first. On each turn Alice (Bob) chooses a (b) uncolored vertices and properly colors them with colors from X. Alice wins if the players eventually create a proper coloring of G; otherwise Bob wins when one of the players has no legal move. The (a,b)-game chromatic number of G, denoted (a,b)-χg(G), is the least integer t such that Alice has a winning strategy when the game is played on G using t colors. We show that if the weak acyclic chromatic number of G is at most k then (2,1)-χg(G)⩽12(k2+3k)