Gašper Fijavž

K 6-Minors in Projective Planar Graphs

It is shown that every 5-connected graph embedded in the projective plane with face-width at least 3 contains the complete graph on 6 vertices as a minor.

On the maximum number of cliques in a graph embedded in a surface

This paper studies the following question: given a surface Σ and an integer n , what is the maximum number of cliques in an n -vertex graph embeddable in Σ ? We characterise the extremal graphs for this question, and prove that the answer is between 8 ( n - ω ) + 2 ω and 8 n + 5 2 2 ω + o ( 2 ω ) , where ω is the maximum integer such that the complete graph K ω embeds in Σ . For the surfaces S 0 , S 1 , S 2 , N 1 , N 2 , N 3 and N 4 we establish an exact answer.

Threshold-coloring and unit-cube contact representation of planar graphs

Abstract In this paper we study threshold-coloring of graphs, where the vertex colors represented by integers are used to describe any spanning subgraph of the given graph as follows. A pair of vertices with a small difference in their colors implies that the edge between them is present, while a pair of vertices with a big color difference implies that the edge is absent. Not all planar graphs are threshold-colorable, but several subclasses, such as trees, some planar grids, and planar graphs with no short cycles can always be threshold-colored. Using these results we obtain unit-cube contact representation of several subclasses of planar graphs. Variants of the threshold-coloring problem are related to well-known graph coloring and other graph-theoretic problems. Using these relations we show the NP-completeness for two of these variants, and describe a polynomial-time algorithm for another.

Minor-minimal 6-regular graphs in the Klein bottle

Let K6 denote the class of all 6-regular graphs which admit an embedding into the Klein bottle. Using the characterization of graphs in K6 we find minor minimal graphs in K6. As a corollary we show that (i) every 6-regular Klein bottlal graph contains a 6-connected minor, and (ii) no 6-regular graph admits an embedding in both the torus and the Klein bottle.

Threshold-Coloring and Unit-Cube Contact Representation of Graphs

We study threshold coloring of graphs where the vertex colors, represented by integers, describe any spanning subgraph of the given graph as follows. Pairs of vertices with near colors imply the edge between them is present and pairs of vertices with far colors imply the edge is absent. Not all planar graphs are threshold-colorable, but several subclasses, such as trees, some planar grids, and planar graphs with no short cycles can always be threshold-colored. Using these results we obtain unit-cube contact representation of several subclasses of planar graphs. We show the NP-completeness for two variants of the threshold coloring problem and describe a polynomial-time algorithm for another.

3-Connected planar graphs are 5-distinguishing colorable with two exceptions

A graph G is said to be d-distinguishing colorable if there is a d -coloring of G such that no automorphism of G except the identity map preserves colors. We shall prove that every 3-connected planar graph is 5-distinguishing colorable except K 2,2,2 and C 6 + overline( K ) 2 and that every 3-connected bipartite planar graph is 3-distinguishing colorable except Q 3 and R ( Q 3 ).

Geometric Realization of Möbius Triangulations

A Mobius triangulation is a triangulation on the Mobius band. A geometric realization of a map

Rigidity and Separation Indices of Graphs in Surfaces

M

Odd complete minors in even embeddings on surfaces

on a surface

Minimum Degree and Graph Minors

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