Július Czap

Learn More
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 has at most n − 2 crossings; moreover, this bound is tight. By this novel necessary condition for 1-planarity, we characterize the 1-planarity of Cartesian product(More)
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, in addition, for each face f and each color c, either no edge or an odd number of edges incident with f is colored with c. It is known that any 2-edgeconnected(More)
A sequence s1, s2, . . . , sk, s1, s2, . . . , sk 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 plane. A facial path consists of consecutive vertices on the boundary of a face. A facial nonrepetitive vertex coloring of(More)
We consider a vertex colouring of a connected plane graph G. A colour c is used k times by a face α of G if it appears k times along the facial walk of α. We prove that every connected plane graph with minimum face degree at least 3 has a vertex colouring with four colours such that every face uses some colour an odd number of times. We conjecture that such(More)
An edge coloring of a graph G is called Mi-edge coloring if at most i colors appear at any vertex of G. Let Ki(G) denote the maximum number of colors used in an Mi-edge coloring of G. In this paper we determine the exact value of K2(G) for any graph G of maximum degree at most 3. Mathematics Subject Classification: 05C15, 05C38
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 coloring, polychromatic coloring, l-facial coloring, nonrepetitive coloring, odd coloring, unique-maximum coloring, WORM coloring, ranking coloring and packing(More)