Journal of Integer Sequences, Vol. 22 (2019), Article 19.2.5 |

School of Computer Science

University of St Andrews

St Andrews, Fife KY16 9SX

United Kingdom

Sergey Kitaev

School of Computer and Information Sciences

University of Strathclyde

Glasgow, G1 1HX

United Kingdom

Hans Zantema

Department of Computer Science

Eindhoven University of Technology

P. O. Box 513

5600 MB Eindhoven

The Netherlands

**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 ∈ 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. Also, we prove that a certain graph has highest representation
number among all comparability graphs on odd number of vertices.

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.

(Concerned with sequences A156808 A290814 A319489 A319490 A319491 A319492.)

Received
October 1 2018; revised versions received February 22 2019; February 23 2019.
Published in *Journal of Integer Sequences*,
February 24 2019.

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