In the distributed message-passing setting a communication network is represented by a graph whose vertices represent processors that perform local computations and communicate over the edges of the graph. In the distributed edge-coloring problem the processors are required to assign colors to edges, such that all edges incident on the same vertex are assigned distinct colors. The previouslykno...

A V-coloring of a graph G is a coloring of the nodes of G with V colors in such a way any two neighboring nodes have different colors. We prove that there exists a O(∆ log n) time distributed algorithm computing a O(∆)-coloring for unit disc graphs under the signal-tointerference-plus-noise ratio (SINR)-based physical model (∆ is the maximum degree of the graph). We also show that, for a well d...

A graph is H-free if it does not contain an induced subgraph isomorphic to the graph H. The graph Pk denotes a path on k vertices. The `-Coloring problem is the problem to decide whether a graph can be colored with at most ` colors such that adjacent vertices receive different colors. We show that 4-Coloring is NP-complete for P8free graphs. This improves a result of Le, Randerath, and Schierme...

Total variant of well known graph coloring game is considered. We determine exact values of χ ′′ g for some classes of graphs and show that total game chromatic number is bounded from above by ∆ + 3∆ +. We also show relation between total game coloring number and game coloring index.