On semi-transitive orientability of triangle-free graphs

An orientation of a graph is semi-transitive if it is acyclic, and for any directed path either there is no arc between and , or is an arc for all . An undirected graph is semi-transitive if it admits a semi-transitive orientation. Semi-transitive graphs generalize several important classes of graphs and they are precisely the class of word-representable graphs studied extensively in the literature. Determining if a triangle-free graph is semi-transitive is an NP-hard problem. The existence of non-semi-transitive triangle-free graphs was established via Erdős' theorem by Halldórsson and the authors in 2011. However, no explicit examples of such graphs were known until recent work of the first author and Saito who have shown computationally that a certain subgraph on 16 vertices of the triangle-free Kneser graph is not semi-transitive, and have raised the question on the existence of smaller triangle-free non-semi-transitive graphs. In this paper we prove that the smallest triangle-free 4-chromatic graph on 11 vertices (the Gr"otzsch graph) and the smallest triangle-free 4-chromatic 4-regular graph on 12 vertices (the Chvátal graph) are not semi-transitive. Hence, the Gr"otzsch graph is the smallest triangle-free non-semi-transitive graph. We also prove the existence of semi-transitive graphs of girth 4 with chromatic number 4 including a small one (the circulant graph on 13 vertices) and dense ones (Toft's graphs). Finally, we show that each -regular circulant graph (possibly containing triangles) is semi-transitive

Radio labeling with pre-assigned frequencies

A radio labeling of a graph $G$ is an assignment of pairwise distinct, positive integer labels to the vertices of $G$ such that labels of adjacent vertices differ by at least $2$. The radio labeling problem (\mbox{\sc RL}) consists in determining a radio labeling that minimizes the maximum label that is used (the so-called span of the labeling). \mbox{\sc RL} is a well-studied problem, mainly motivated by frequency assignment problems in which transmitters are not allowed to operate on the same frequency channel. We consider the special case where some of the transmitters have pre-assigned operating frequency channels. This leads to the natural variants \mbox{\sc p-RL($l$)} and \mbox{\sc p-RL($*$)} of \mbox{\sc RL} with $l$ pre-assigned labels and an arbitrary number of pre-assigned labels, respectively

REPORTS IN INFORMATICS

Finding minimum feedback vertex set in bipartite grap

On avoidance of V- and Λ-patterns in permutations

We study V- and Λ-patterns which generalize valleys and peaks, as well as increasing and decreasing runs, in permutations. A complete classification of permutations (multi)-avoiding V- and Λ-patterns of length 4 is given. We also establish a connection between restricted permutations and matchings in the coronas of complete graphs

Finding a minimum feedback vertex set in time O(1.7548 n

Abstract. We present an O(1.7548 n) algorithm finding a minimum feedback vertex set in a graph on n vertices

On 4-chromatic edge-critical regular graphs of high connectivity

AbstractNew examples of 4-chromatic edge-critical r-regular and r-connected graphs are presented for r=6,8,10