2025/01/23 by Mohammed B. Alshammari, Sergey Kitaev, Alshammari, Mohammed +7
Computer Science · #05C62 #Advanced Graph Theory Research #Combinatorics (math.CO) #Constraint Satisfaction and Optimization #FOS: Mathematics #semigroups and automata theory
paper · doi:10.48550/arxiv.2501.13871
openalex publication_date 2025/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.