Stanislav Jendrol' | Searcharxiv

arXiv2020

Graph polynomials and paintability of plane graphs

There exists a variety of coloring problems for plane graphs, involving vertices, edges, and faces in all possible combinations. For instance, in the \emph{entire coloring} of a plane graph we are to color these three sets so that any pair of adjacent or incident elements get different colors. We study here some problems of this type from algebraic perspective, focusing on the \emph{facial} variant. We obtain several results concerning the \emph{Alon-Tarsi number} of various graphs derived from plane embeddings. This allows for extensions of some previous results for \emph{choosability} of these graphs to the game theoretic variant, know as \emph{paintability}. For instance, we prove that every plane graph is facially entirely \emph{$8$-paintable}, which means (metaphorically) that even a color-blind person can facially color the entire graph form lists of size $8$.

math.CO↗

arXiv2018

Graphs with conflict-free connection number two

An edge-colored graph $G$ is \emph{conflict-free connected} if any two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The \emph{conflict-free connection number} of a connected graph $G$, denoted by $cfc(G)$, is the smallest number of colors needed in order to make $G$ conflict-free connected. For a graph $G,$ let $C(G)$ be the subgraph of $G$ induced by its set of cut-edges. In this paper, we first show that, if $G$ is a connected non-complete graph $G$ of order $n\geq 9$ with $C(G)$ being a linear forest and with the minimum degree %$δ(G)\geq 2$, then $cfc(G)=2$ for $4 \leq n\leq 8 $; if $δ(G)\geq \max\{3, \frac{n-4}{5}\}$, then $cfc(G)=2$. The bound on the minimum degree is best possible. Next, we prove that, if $G$ is a connected non-complete graph of order $n\geq 33$ with $C(G)$ being a linear forest and with $d(x)+d(y)\geq \frac{2n-9}{5}$ for each pair of two nonadjacent vertices $x, y$ of $V(G)$, then $cfc(G)=2$. Both bounds, on the order $n$ and the degree sum, are tight. Moreover, we prove several results concerning relations between degree conditions on $G$ and the number of cut edges in $G$.

math.CO↗