arXiv subjects

Riste Škrekovski

Publications and source records attributed to Riste Škrekovski.

76 records · Page 5Linked to original sources

arXiv2013

Improved bound on facial parity edge coloring

A facial parity edge coloring of a 2-edge connected plane graph is an edge coloring where no two consecutive edges of a facial walk of any face receive the same color. Additionally, for every face f and every color c either no edge or an odd number of edges incident to f are colored by c. Czap, Jendrol', Kardoš and Sotak showed that every 2-edge connected plane graph admits a facial parity edge coloring with at most 20 colors. We improve this bound to 16 colors.

math.CO↗

arXiv2013

l-facial edge colorings of graphs

An l-facial edge coloring of a plane graph is a coloring of the edges such that any two edges at distance at most l on a boundary walk of some face receive distinct colors. It is conjectured that 3l + 1 colors suffice for an l-facial edge coloring of any plane graph. We prove that 7 colors suffice for a 2-facial edge coloring of any plane graph and therefore confirm the conjecture for l = 2.

math.CO↗

arXiv2013

Euler's idoneal numbers and an inequality concerning minimal graphs with a prescribed number of spanning trees

Let $α(n)$ be the least number $k$ for which there exists a simple graph with $k$ vertices having precisely $n \geq 3$ spanning trees. Similarly, define $β(n)$ as the least number $k$ for which there exists a simple graph with $k$ edges having precisely $n \geq 3$ spanning trees. As an $n$-cycle has exactly $n$ spanning trees, it follows that $α(n),β(n) \leq n$. In this paper, we show that $α(n) \leq \frac{n+4}{3}$ and $β(n) \leq \frac{n+7}{3} $ if and only if $n \notin {3,4,5,6,7,9,10,13,18,22}$, which is a subset of Euler's idoneal numbers. Moreover, if $n \not \equiv 2 \pmod{3}$ and $n \not = 25$ we show that $α(n) \leq \frac{n+9}{4}$ and $β(n) \leq \frac{n+13}{4}.$ This improves some previously known bounds.

math.CO↗

arXiv2011

Graphs with two crossings are 5-choosable

A graph G is k-choosable if G can be properly colored whenever every vertex has a list of at least k available colors. Thomassen's theorem states that every planar graph is 5-choosable. We extend the result by showing that every graph with at most two crossings is 5-choosable.

math.CO↗