New Tools to Study 1-11-Representation of Graphs

Abstract

The notion of a k-11-representable graph was introduced by Jeff Remmel in 2017 and studied by Cheon et al. in 2019 as a natural extension of the extensively studied notion of word-representable graphs, which are precisely 0-11-representable graphs. A graph G is k-11-representable if it can be represented by a word w such that for any edge (resp., non-edge) xy in G the subsequence of w formed by x and y contains at most k (resp., at least k+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k+1$$\\end{document}) pairs of consecutive equal letters. A remarkable result of Cheon at al. is that any graph is 2-11-representable, while it is unknown whether every graph is 1-11-representable. Cheon et al. showed that the class of 1-11-representable graphs is strictly larger than that of word-representable graphs, and they introduced a useful toolbox to study 1-11-representable graphs. In this paper, we introduce new tools for studying 1-11-representation of graphs. We apply them for establishing 1-11-representation of Chvátal graph, Mycielski graph, split graphs, and graphs whose vertices can be partitioned into a comparability graph and an independent set.