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### Decompositions of Plane Graphs Under Parity Constrains Given by Faces

Discussiones Mathematicae Graph Theory

An edge coloring of a plane graph G is facially proper if no two faceadjacent edges of G receive the same color. A facial (facially proper) parity edge coloring of a plane graph G is an (facially proper) edge coloring with the property that, for each color c and each face f of G, either an odd number of edges incident with f is colored with c, or color c does not occur on the edges of f. In this paper we deal with the following question: For which integers k does there exist a facial (facially proper)...

### Fractional Q-Edge-Coloring of Graphs

Discussiones Mathematicae Graph Theory

An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let [...] be an additive hereditary property of graphs. A [...] -edge-coloring of a simple graph is an edge coloring in which the edges colored with the same color induce a subgraph of property [...] . In this paper we present some results on fractional [...] -edge-colorings. We determine the fractional [...] -edge chromatic number for matroidal properties of graphs.

### Colouring vertices of plane graphs under restrictions given by faces

Discussiones Mathematicae Graph Theory

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...

### On the strong parity chromatic number

Discussiones Mathematicae Graph Theory

A vertex colouring of a 2-connected plane graph G is a strong parity vertex colouring if for every face f and each colour c, the number of vertices incident with f coloured by c is either zero or odd. Czap et al. in [9] proved that every 2-connected plane graph has a proper strong parity vertex colouring with at most 118 colours. In this paper we improve this upper bound for some classes of plane graphs.

### M 2 -Edge Colorings Of Cacti And Graph Joins

Discussiones Mathematicae Graph Theory

An edge coloring φ of a graph G is called an M2-edge coloring if |φ(v)| ≤ 2 for every vertex v of G, where φ(v) is the set of colors of edges incident with v. Let 𝒦2(G) denote the maximum number of colors used in an M2-edge coloring of G. In this paper we determine 𝒦2(G) for trees, cacti, complete multipartite graphs and graph joins.

### On An Extremal Problem In The Class Of Bipartite 1-Planar Graphs

Discussiones Mathematicae Graph Theory

A graph G = (V, E) 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 bipartite 1-planar graphs with prescribed numbers of vertices in partite sets. Bipartite 1-planar graphs are known to have at most 3n − 8 edges, where n denotes the order of a graph. We show that maximal-size bipartite 1-planar graphs which are almost balanced have not significantly fewer edges than indicated by this upper bound, while the same is not true...

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