Anders Sune Pedersen

Independence, odd girth, and average degree

We prove several tight lower bounds in terms of the order and the average degree for the independence number of graphs that are connected and/or satisfy some odd girth condition. Our main result is the extension of a lower bound for the independence number of triangle-free graphs of maximum degree at most three due to Heckman and Thomas [Discrete Math 233 (2001), 233–237] to arbitrary triangle-free graphs. For connected triangle-free graphs of order n and size m, our result implies the existence of an independent set of order at least (4n−m−1)/7.

Lower bounds on the independence number of certain graphs of odd girth at least seven

Heckman and Thomas [C.C. Heckman, R. Thomas, A new proof of the independence ratio of triangle-free cubic graphs, Discrete Math. 233 (2001) 233-237] proved that every connected subcubic triangle-free graph G has an independent set of order at least (4n(G)-m(G)-1)/7 where n(G) and m(G) denote the order and size of G, respectively. We conjecture that every connected subcubic graph G of odd girth at least seven has an independent set of order at least (5n(G)-m(G)-1)/9 and verify our conjecture under some additional technical assumptions.

Hadwiger's Conjecture and inflations of the Petersen graph

An inflation of a graph G is obtained by replacing vertices in G by disjoint cliques and adding all possible edges between any pair of cliques corresponding to adjacent vertices in G . We prove that the chromatic number of an arbitrary inflation F of the Petersen graph is equal to the chromatic number of some inflated 5-cycle contained in F . As a corollary, we find that Hadwigers Conjecture holds for any inflation of the Petersen graph. This solves a problem posed by Bjarne Toft.

Recolouring-resistant colourings

We study colourings of graphs with the property that the number of used colours cannot be reduced by applying some recolouring operation. A well-studied example of such colourings are b-colourings, which were introduced by Irving and Manlove [R.W. Irving, D.F. Manlove, The b-chromatic number of a graph, Discrete Appl. Math. 91 (1999) 127-141]. Given a graph and a colouring, a recolouring operation specifies a set of vertices of the graph on which the colouring can be changed. We consider two such operations: One which allows the recolouring of all vertices within some given distance of some colour class, and another which allows the recolouring of all vertices that belong to one of a given number of colour classes. Our results extend known results concerning b-colourings and the associated b-chromatic number.

A basic elementary extension of the Duchet-Meyniel theorem

The Conjecture of Hadwiger implies that the Hadwiger number h times the independence number @a of a graph is at least the number of vertices n of the graph. In 1982 Duchet and Meyniel [P. Duchet, H. Meyniel, On Hadwigers number and the stability number, Ann. of Discrete Math. 13 (1982) 71-74] proved a weak version of the inequality, replacing the independence number @a by 2@a-1, that is, (2@a-1)@?h>=n. In 2005 Kawarabayashi, Plummer and the second author [K. Kawarabayashi, M. Plummer, B. Toft, Improvements of the theorem of Duchet and Meyniel on Hadwigers Conjecture, J. Combinatorial Theory B 95 (2005) 152-167] published an improvement of the theorem, replacing 2@a-1 by 2@a-3/2 when @a is at least 3. Since then a further improvement by Kawarabayashi and Song has been obtained, replacing 2@a-1 by 2@a-2 when @a is at least 3. In this paper a basic elementary extension of the Theorem of Duchet and Meyniel is presented. This may be of help to avoid dealing with basic cases when looking for more substantial improvements. The main unsolved problem (due to Seymour) is to improve, even just slightly, the theorem of Duchet and Meyniel in the case when the independence number @a is equal to 2. The case @a=2 of Hadwigers Conjecture was first pointed out by Mader as an interesting special case.