arXiv subjects

Robert Šámal

Publications and source records attributed to Robert Šámal.

39 records · Page 3Linked to original sources

arXiv2012

Cycle-continuous mappings -- order structure

Given two graphs, a mapping between their edge-sets is cycle-continuous, if the preimage of every cycle is a cycle. The motivation for this notion is Jaeger's conjecture that for every bridgeless graph there is a cycle-continuous mapping to the Petersen graph. Answering a question of DeVos, Nešetřil, and Raspaud, we prove that there exists an infinite set of graphs with no cycle-continuous mapping between them. Further extending this result, we show that every countable poset can be represented by graphs and existence of cycle-continuous mappings between them.

math.CO↗

arXiv2011

Star chromatic index

The star chromatic index $χ_s'(G)$ of a graph $G$ is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored. We obtain a near-linear upper bound in terms of the maximum degree $Δ=Δ(G)$. Our best lower bound on $χ_s'$ in terms of $Δ$ is $2Δ(1+o(1))$ valid for complete graphs. We also consider the special case of cubic graphs, for which we show that the star chromatic index lies between 4 and 7 and characterize the graphs attaining the lower bound. The proofs involve a variety of notions from other branches of mathematics and may therefore be of certain independent interest.

math.CO↗

arXiv2010

An Eberhard-like theorem for pentagons and heptagons

Eberhard proved that for every sequence $(p_k), 3\le k\le r, k\ne 5,7$ of non-negative integers satisfying Euler's formula $\sum_{k\ge3} (6-k) p_k = 12$, there are infinitely many values $p_6$ such that there exists a simple convex polyhedron having precisely $p_k$ faces of length $k$ for every $k\ge3$, where $p_k=0$ if $k>r$. In this paper we prove a similar statement when non-negative integers $p_k$ are given for $3\le k\le r$, except for $k=5$ and $k=7$. We prove that there are infinitely many values $p_5,p_7$ such that there exists a simple convex polyhedron having precisely $p_k$ faces of length $k$ for every $k\ge3$. %, where $p_k=0$ if $k>r$. We derive an extension to arbitrary closed surfaces, yielding maps of arbitrarily high face-width. Our proof suggests a general method for obtaining results of this kind.

math.CO↗