arXiv subjects

Roman Soták

Publications and source records attributed to Roman Soták.

23 records · Page 2Linked to original sources

arXiv2016

On incidence coloring conjecture in Cartesian products of graphs

An incidence in a graph $G$ is a pair $(v,e)$ where $v$ is a vertex of $G$ and $e$ is an edge of $G$ incident to $v$. Two incidences $(v,e)$ and $(u,f)$ are adjacent if at least one of the following holds: $(a)$ $v = u$, $(b)$ $e = f$, or $(c)$ $vu \in \{e,f\}$. An incidence coloring of $G$ is a coloring of its incidences assigning distinct colors to adjacent incidences. It was conjectured that at most $Δ(G) + 2$ colors are needed for an incidence coloring of any graph $G$. The conjecture is false in general, but the bound holds for many classes of graphs. We introduce some sufficient properties of the two factor graphs of a Cartesian product graph $G$ for which $G$ admits an incidence coloring with at most $Δ(G) + 2$ colors.

math.CO↗

arXiv2013

Strong edge coloring of subcubic bipartite graphs

A strong edge coloring of a graph $G$ is a proper edge coloring in which each color class is an induced matching of $G$. In 1993, Brualdi and Quinn Massey proposed a conjecture that every bipartite graph without $4$-cycles and with the maximum degrees of the two partite sets $2$ and $Δ$ admits a strong edge coloring with at most $Δ+2$ colors. We prove that this conjecture holds for such graphs with $Δ=3$. We also confirm the conjecture proposed by Faudree et al. for subcubic bipartite graphs.

math.CO↗

arXiv2013

Star edge coloring of some classes of graphs

\textit{A star edge coloring} of a graph is a proper edge coloring without bichromatic paths and cycles of length four. In this paper we establish tight upper bounds for trees and subcubic outerplanar graphs, and derive an upper bound for outerplanar graphs.

math.CO↗

arXiv2013

Strong edge coloring of planar graphs

A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most two receive distinct colors. It is known that every planar graph with maximum degree D has a strong edge coloring with at most 4D + 4 colors. We show that 3D + 6 colors suffice if the graph has girth 6, and 3D colors suffice if the girth is at least 7. Moreover, we show that cubic planar graphs with girth at least 6 can be strongly edge colored with at most 9 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↗