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**public sources and our publisher partners.**Abstract In this survey the following types of colorings of plane graphs are discussed, both in their vertex and edge versions: facially proper coloring, rainbow coloring, antirainbow coloring, loose… Continue Reading

A graph is called 1-planar if it can be drawn in the plane so that each of its edges is crossed by at most one other edge. We show that every 1-planar drawing of any 1-planar graph on $n$ vertices… Continue Reading

A sequence is a repetition. A sequence S is nonrepetitive, if no subsequence of consecutive terms of S is a repetition. Let G be a plane graph. That is, a planar graph with a fixed embedding in the… Continue Reading

A graph is called 1-planar if there exists its drawing in the plane such that each edge is crossed at most once. We present the full characterization of 1-planar complete k-partite graphs.

A facial parity edge coloring of a 2-edge-connected plane graph is such an edge coloring in which no two face-adjacent edges (consecutive edges of a facial walk of some face) receive the same color,… Continue Reading

A graph is called 1-planar if it can be drawn in the plane so that each of its edges is crossed by at most one other edge. In 2014, Zhang showed that the set of all 1-planar graphs can be decomposed… Continue Reading

Abstract A facial parity edge colouring of a connected bridgeless plane graph is such an edge colouring in which no two face-adjacent edges receive the same colour and, in addition, for each face f… Continue Reading

A 1-planar graph is a graph that can be drawn in the plane such that each edge is crossed by at most one other edge. In this paper we give an upper bound for the total chromatic number for 1-planar… Continue Reading

Abstract Given three planar graphs F,H, and G, an (F,H)-WORM coloring of G is a vertex coloring such that no subgraph isomorphic to F is rainbow and no subgraph isomorphic to H is monochromatic. If G… Continue Reading

A graph is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. In this paper, we study 1-planar lexicographic products. Mathematics Subject… Continue Reading