DP-coloring, also known as correspondence coloring, is introduced by Dvořák and Postle. It is a generalization of list coloring. In this paper, we show that every connected toroidal graph without triangles adjacent to 5-cycles has minimum degree at most three unless it is a 2-connected 4-regular graph with Euler characteristic (G) = 0. Consequently, every toroidal graph without triangles adjace...

A graph G is uniquely k-colorable if the chromatic number of G is k and G has only one k-coloring up to permutation of the colors. A uniquely k-colorable graph G is edge-critical if G−e is not a uniquely k-colorable graph for any edge e ∈ E(G). In this paper, we prove that if G is an edge-critical uniquely 3-colorable planar graph, then |E(G)| 6 83 |V (G)| − 17 3 . On the other hand, there exis...

An edge-coloring of a graph G with colors 1, . . . , t is an interval t-coloring if all colors are used, and the colors of edges incident to each vertex of G are distinct and form an interval of integers. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. The problem of deciding whether a bipartite graph is interval colorable is NP-complete. The smalle...

Hadwiger's conjecture states that every Kt-minor free graph is (t?1)-colorable. A qualitative strengthening of this raised by Gerards and Seymour, known as the Odd conjecture, similarly with no odd For both conjectures, their asymptotic relaxations remain open, i.e., whether an upper bound on chromatic number form Ct for some constant C>0 exists. We show if without a f(t)-colorable, then 2f(t)-...

The \emph{slow-coloring game} is played by Lister and Painter on a graph $G$. Initially, all vertices of $G$ are uncolored. In each round, marks nonempty set $M$ uncolored vertices, colors subset that independent in game ends when colored. score the sum sizes sets marked Lister. goal to minimize score, while tries maximize it. We provide strategies for various classes graphs whose can be partit...

The problems under consideration here were united for the first time by the general term “The Three Color Problem” in Ore’s book [7], where the author was interested in the description of planar 3-colorable graphs. Our graph theoretic terminology is that of Harary [4], except that we use vertices and edges instead of points and lines, respectively. The graphs considered here are planar and have...