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Edge homogeneous colorings

Madaras Tomáš, Onderko Alfréd, Schweser Thomas

Opuscula Mathematica

2022

Vol. 42, no. 1

65--73

We explore four kinds of edge colorings defined by the requirement of equal number of colors appearing, in particular ways, around each vertex or each edge. We obtain the characterization of graphs colorable in such a way that the ends of each edge see (not regarding the edge color itself) q colors (resp. one end sees q colors and the color sets for both ends are the same), and a sufficient condition for 2-coloring a graph in a way that the ends of each edge see (with the omission of that edge color) altogether q colors. The relations of these colorings to Mq-colorings and role colorings are also discussed; we prove an interpolation theorem for the numbers of colors in edge coloring where all edges around each vertex have q colors.

Line graphs, their Desarguesian closures, and corresponding groups of automorphisms

Jankowska B., Prażmowska M., Prażmowski K.

Demonstratio Mathematica

2007

Vol. 40, nr 4

971-986

The notion of Desarguesian closure of an arbitrary graph was introduced in [7], and basic properties of Desarguesian closure of complete graphs were also presented in [7]. Then, in [4], the Desarguesian closure of binomial graphs (cf. [5]) was studied. In this paper we shall be mainly concerned with the line graphs associated with complete graphs, their Desarguesian closure, horizon, and automorphisms.