On Semi-transitive Orientations and Graphs Representable by Words

On Representable Graphs, Semi-transitive Orientations, and the Representation Numbers

A graph G = (V,E) is representable if there exists a word W over the alphabet V such that letters x and y alternate in W if and only if (x, y) ∈ E for each x 6= y. If W is k-uniform (each letter of W occurs exactly k times in it) then G is called k-representable. It was shown in [4] that a graph is representable if and only if it is k-representable for some k. Minimum k for which a representabl...

Semi-transitive orientations and word-representable graphs

Completing orientations of partially oriented graphs

We initiate a general study of what we call orientation completion problems. For a fixed class C of oriented graphs, the orientation completion problem asks whether a given partially oriented graph P can be completed to an oriented graph in C by orienting the (non-oriented) edges in P . Orientation completion problems commonly generalize several existing problems including recognition of certai...

Word-representability of triangulations of grid-covered cylinder graphs

A graph G = (V,E) is word-representable if there exists a word w over the alphabet V such that letters x and y, x 6= y, alternate in w if and only if (x, y) ∈ E. Halldórsson, Kitaev and Pyatkin have shown that a graph is word-representable if and only if it admits a so-called semi-transitive orientation. A corollary of this result is that any 3-colorable graph is wordrepresentable.Masárová Akro...

Product of normal edge-transitive Cayley graphs

For two normal edge-transitive Cayley graphs on groups H and K which have no common direct factor and $gcd(|H/H^prime|,|Z(K)|)=1=gcd(|K/K^prime|,|Z(H)|)$, we consider four standard products of them and it is proved that only tensor product of factors can be normal edge-transitive.