Július Czap

1-planarity of complete multipartite graphs

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.

Facial Nonrepetitive Vertex Coloring of Plane Graphs

A sequence s1, s2, . . . , sk, s1, s2, . . . , sk is a repetition. A sequence S is nonrepetitive, if no subsequence of consecutive terms of S form a repetition. Let G be a vertex colored graph. A path of G is nonrepetitive, if the sequence of colors on its vertices is nonrepetitive. If G is a plane graph, then a facial nonrepetitive vertex coloring of G is a vertex coloring such that any facial path is nonrepetitive. Let πf (G) denote the minimum number of colors of a facial nonrepetitive vertex coloring of G. Jendroľ and Harant posed a conjecture that πf (G) can be bounded from above by a constant. We prove that πf (G) ≤ 24 for any plane graph G.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 plane. A facial path consists of consecutive vertices on the boundary of a face. A facial nonrepetitive vertex coloring of a plane graph G is a vertex coloring such that the colors assigned to the vertices of any facial path form a nonrepetitive sequence. Let denote the minimum number of colors of a facial nonrepetitive vertex coloring of G. Harant and Jendrol’ conjectured that can be bounded from above by a constant. We prove that for any plane graph G.

Colouring vertices of plane graphs under restrictions given by faces

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 a colouring can be done using three colours. We prove that this conjecture is true for 2-connected cubic plane graphs. Next we consider other three kinds of colourings that require stronger restrictions.

Vertex coloring of plane graphs with nonrepetitive boundary paths

A sequence s1, s2, . . . , sk, s1, s2, . . . , sk is a repetition. A sequence S is nonrepetitive, if no subsequence of consecutive terms of S form a repetition. Let G be a vertex colored graph. A path of G is nonrepetitive, if the sequence of colors on its vertices is nonrepetitive. If G is a plane graph, then a facial nonrepetitive vertex coloring of G is a vertex coloring such that any facial path is nonrepetitive. Let πf (G) denote the minimum number of colors of a facial nonrepetitive vertex coloring of G. Jendroľ and Harant posed a conjecture that πf (G) can be bounded from above by a constant. We prove that πf (G) ≤ 24 for any plane graph G.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 plane. A facial path consists of consecutive vertices on the boundary of a face. A facial nonrepetitive vertex coloring of a plane graph G is a vertex coloring such that the colors assigned to the vertices of any facial path form a nonrepetitive sequence. Let denote the minimum number of colors of a facial nonrepetitive vertex coloring of G. Harant and Jendrol’ conjectured that can be bounded from above by a constant. We prove that for any plane graph G.

Facial parity edge colouring of plane pseudographs

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 and each colour c , either no edge or an odd number of edges incident with f is coloured with c . Let χ p ′ ( G ) denote the minimum number of colours used in such a colouring of G . In this paper we prove that χ p ′ ( G ) ≤ 20 for any 2-edge-connected plane graph G . In the case when G is a 3 -edge-connected plane graph the upper bound for this parameter is 12 . For G being 4 -edge-connected plane graph we have χ p ′ ( G ) ≤ 9 . On the other hand we prove that some bridgeless plane graphs require at least 10 colours for such a colouring.

Facial parity edge colouring

A facial parity edge colouring of a connected bridgeless plane graph is an edge colouring in which no two face-adjacent edges (consecutive edges of a facial walk of some face) receive the same colour, in addition, for each face α and each colour c , either no edge or an odd number of edges incident with \alpha is coloured with c . From Vizings theorem it follows that every 3-connected plane graph has a such colouring with at most Δ * + 1 colours, where Δ * is the size of the largest face. In this paper we prove that any connected bridgeless plane graph has a facial parity edge colouring with at most 92 colours.

Facially-constrained colorings of plane graphs: A survey

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 coloring, polychromatic coloring, l -facial coloring, nonrepetitive coloring, odd coloring, unique-maximum coloring, WORM coloring, ranking coloring and packing coloring. In the last section of this paper we show that using the language of words these different types of colorings can be formulated in a more general unified setting.

Three classes of 1-planar graphs

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 this paper we decompose the set of all 1-planar graphs into three classes C0, C1 and C2 with respect to the types of crossings and present the decomposition of 1-planar join products. Zhang [8] proved that every n-vertex 1-planar graph of class C1 has at most 18 5 n − 36 5 edges and a C1-drawing with at most 3 5 n− 6 5 crossings. We improve these results. We show that every C1-drawing of a 1-planar graph has at most 3 5 n − 6 5 crossings. Consequently, every n-vertex 1-planar graph of class C1 has at most 18 5 n− 36 5 edges. Moreover, we prove that this bound is sharp.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 into three classes

Parity vertex coloring of outerplane graphs

\mathcal C_0, \mathcal C_1

Facial packing edge-coloring of plane graphs

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