A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, in such a way that no two adjacent or incident elements receive the same color. The total coloring problem is to find a total coloring of a given graph with the minimum number of colors. Many combinatorial problems can be efficiently solved for partial k-trees, i.e., graphs with bounded tree-width. Howeve...

We study routing and path coloring problems in all-optical networks as non-cooperative games. We especially focus on oblivious payment functions, that is, functions that charge a player according to her own strategy only. We first strengthen a known relation between such games and online routing and path coloring. In particular, we show that the price of anarchy of such games is lower-bounded b...

An odd c-coloring of a graph is proper such that each non-isolated vertex has color appearing an number times on its neighborhood. This concept was introduced very recently by Petruševski and Škrekovski attracted considerable attention. Cranston investigated colorings graphs with bounded maximum average degree, conjectured every G mad(G)≤4c−4c+1 for c≥4, proved the conjecture c∈{5,6}. In partic...

DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvořák and Postle. We introduce study (i,j)-defective DP-colorings multigraphs. concentrate on sparse multigraphs consider fDP(i,j,n) — the minimum number edges that may have an n-vertex (i,j)-critical multigraph, is, multigraph G has no but whose every proper subgraph such coloring. F...