Word-Representability of Line Graphs

Abstract

A graph <i>G</i>=(<i>V,E</i>) is representable if there exists a word <i>W</i> over the alphabet <i>V</i> such that letters <i>x</i> and <i>y</i> alternate in <i>W</i> if and only if (<i>x</i> ,<i>y</i>) is in <i>E</i> for each <i>x</i> not equal to <i>y</i> . The motivation to study representable graphs came from algebra, but this subject is interesting from graph theoretical, computer science, and combinatorics on words points of view. In this paper, we prove that for <i>n</i> greater than 3, the line graph of an <i>n</i>-wheel is non-representable. This not only provides a new construction of non-repre- sentable graphs, but also answers an open question on representability of the line graph of the 5-wheel, the minimal non-representable graph. Moreover, we show that for <i>n</i> greater than 4, the line graph of the complete graph is also non-representable. We then use these facts to prove that given a graph <i>G</i> which is not a cycle, a path or a claw graph, the graph obtained by taking the line graph of <i>G k</i>-times is guaranteed to be non-representable for <i>k</i> greater than 3

Word-representability of triangulations of grid-covered cylinder graphs

“…V m−1 and V 0 ). For example, referring to Figure 2.4, the layer L 2 is the induced subgraph formed by the vertices {v 10 , v 11 , v 12 , v 13 , v 20 , v 21 , v 22 , v 23 }, while the sector S 3 is the induced subgraph formed by the vertices {v 02 , v 12 , v 22 , v 03 , v 13 , v 23 }.…”

On word-representability of polyomino triangulations

“…For example, the cycle graph on 4 vertices labeled by 1, 2, 3 and 4 in clockwise direction can be represented by the word 14213243. There is a long line of research on word-representable graphs [1]- [8], and the current paper is a continuation of this line of research. A comprehensive introduction to the theory of word-representable graphs will be given in [5].…”

Word-Representability of Face Subdivisions of Triangular Grid Graphs

“…Recently, a number of (fundamental) results on word-representable graphs were obtained in the literature; for example, see [1], [3], [5], [7], [9], [11], and [12]. In particular, Halldórsson et al [7] have shown that a graph is wordrepresentable if and only if it admits a semi-transitive orientation (to be defined in Section 2), which, among other important corollaries, implies that all 3-colorable graphs are word-representable.…”