On 1-11-representability and multi-1-11-representability of graphs
Mohammed Alshammari, Sergey Kitaev, Chaoliang Tang, Tianyi Tao, Junchi Zhang
TL;DR
The paper investigates $1$-$11$-representability and its multi-variant for graphs, building on the notion that word-representable graphs correspond to $0$-$11$-representability and that every graph is known to be $2$-$11$-representable. It establishes that all graphs with at most eight vertices are $1$-$11$-representable, using a chromatic-number–driven case analysis and a toolbox of operations (such as star and matching insertions) to maintain semi-transitive orientations and thus word-representability. It then generalizes to multi-$1$-$11$-representation numbers, proving that any graph on at most 24 vertices has a strict multi-$1$-$11$-representation number at most $2$ by decomposing the graph into eight-vertex blocks and leveraging existing $1$-$11$-representability results. The work advances both the theoretical understanding of $1$-$11$-representability and practical techniques for constructing representations, with several open questions about higher-colourable classes and tighter bounds for multi-representation numbers.
Abstract
Jeff Remmel introduced the concept of a $k$-11-representable graph in 2017. This concept was first explored by Cheon et al. in 2019, who considered it as a natural extension of word-representable graphs, which are exactly 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 $k+1$) pairs of consecutive equal letters. A remarkable result of Cheon at al. is that any graph is 2-11-representable, while it is still 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, which was extended by additional powerful tools suggested by Futorny et al. in 2024. In this paper, we prove that all graphs on at most 8 vertices are 1-11-representable hence extending the known fact that all graphs on at most 7 vertices are 1-11-representable. Also, we discuss applications of our main result in the study of multi-1-11-representation of graphs we introduce in this paper analogously to the notion of multi-word-representation of graphs suggested by Kenkireth and Malhotra in 2023.