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Solving computational problems in the theory of word-representable graphs
- Source :
- Journal of Integer Sequences, Vol. 22 (2019), Article 19.2.5
- Publication Year :
- 2018
-
Abstract
- A simple graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ iff $xy\in E$. Word-representable graphs generalize several important classes of graphs. A graph is word-representable iff it admits a semi-transitive orientation. We use semi-transitive orientations to enumerate connected non-word-representable graphs up to the size of 11 vertices, which led to a correction of a published result. Obtaining the enumeration results took 3 CPU years of computation. Also, a graph is word-representable iff it is $k$-representable for some $k$, that is, if it can be represented using $k$ copies of each letter. The minimum such $k$ for a given graph is called graph's representation number. Our computational results in this paper not only include distribution of $k$-representable graphs on at most 9 vertices, but also have relevance to a known conjecture on these graphs. In particular, we find a new graph on 9 vertices with high representation number. Finally, we introduce the notion of a $k$-semi-transitive orientation refining the notion of a semi-transitive orientation, and show computationally that the refinement is not equivalent to the original definition unlike the equivalence of $k$-representability and word-representability.
- Subjects :
- Mathematics - Combinatorics
Subjects
Details
- Database :
- arXiv
- Journal :
- Journal of Integer Sequences, Vol. 22 (2019), Article 19.2.5
- Publication Type :
- Report
- Accession number :
- edsarx.1808.01215
- Document Type :
- Working Paper