Henning Ulfarsson

On the representability of line graphs

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 ≠ y. Such a W is called a word-representant of G. Note that in this paper we use the term graph to mean a finite, simple graph, even though the definition of representable is applicable to more general graphs.

Word-Representability of Line Graphs

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) is in E for each x not equal to y . 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 n greater than 3, the line graph of an n-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 n 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 G which is not a cycle, a path or a claw graph, the graph obtained by taking the line graph of G k-times is guaranteed to be non-representable for k greater than 3.

Restricted non-separable planar maps and some pattern avoiding permutations

Tutte founded the theory of enumeration of planar maps in a series of papers in the 1960s. Rooted non-separable planar maps are in bijection with West-2-stack-sortable permutations, @b(1,0)-trees introduced by Cori, Jacquard and Schaeffer in 1997, as well as a family of permutations defined by the avoidance of two four letter patterns. In this paper we study how certain structures in planar maps transfer to trees and permutations via the bijections. More precisely, we show that the number of 2-faces in a map equals the number of nodes in the corresponding @b(1,0)-tree that are single children with maximum label; give upper and lower bounds on the number of multiple-edge-free rooted non-separable planar maps. We also use the bijection between rooted non-separable planar maps and a certain class of permutations, found by Claesson, Kitaev and Steingrimsson in 2009, to show that 2-face-free maps correspond to permutations avoiding certain mesh patterns. Finally, we give asymptotics for some of our enumerative results.