Tommy R. Jensen

Highly Connected Sets and the Excluded Grid Theorem

We present a short proof of the excluded grid theorem of Robertson and Seymour, the fact that a graph has no large grid minor if and only if it has small tree-width. We further propose a very simple obstruction to small tree-width inspired by that proof, showing that a graph has small tree-width if and only if it contains no large highly connected set of vertices.

Describing 3-paths in normal plane maps

Abstract We prove that every normal plane map, as well as every 3-polytope, has a path on three vertices whose degrees are bounded from above by one of the following triplets: ( 3 , 3 , ∞ ) , ( 3 , 4 , 11 ) , ( 3 , 7 , 5 ) , ( 3 , 10 , 4 ) , ( 3 , 15 , 3 ) , ( 4 , 4 , 9 ) , ( 6 , 4 , 8 ) , ( 7 , 4 , 7 ) , and ( 6 , 5 , 6 ) . No parameter of this description can be improved, as shown by appropriate 3-polytopes.

5-Stars of Low Weight in Normal Plane Maps with Minimum Degree 5

Abstract It is known that there are normal plane maps M5 with minimum degree 5 such that the minimum degree-sum w(S5) of 5-stars at 5-vertices is arbitrarily large. In 1940, Lebesgue showed that if an M5 has no 4-stars of cyclic type (5, 6, 6, 5) centered at 5-vertices, then w(S5) ≤ 68. We improve this bound of 68 to 55 and give a construction of a (5, 6, 6, 5)-free M5 with w(S5) = 48

The color space of a graph

If k is a prime power, and G is a graph with n vertices, then a k-coloring of G may be considered as a vector in GF(k)n. We prove that the subspace of GF(3)n spanned by all 3-colorings of a planar triangle-free graph with n vertices has dimension n. In particular, any such graph has at least n - 1 nonequivalent 3-colorings, and the addition of any edge or any vertex of degree 3 results in a 3-colorable graph.

Dense critical and vertex-critical graphs

Abstract This paper gives new constructions of k -chromatic critical graphs with high minimum degree and high edge density, and of vertex-critical graphs with high edge density.

A step towards the strong version of Havel's three color conjecture

In 1970, Havel asked if each planar graph with the minimum distance, d^@?, between triangles large enough is 3-colorable. There are 4-chromatic planar graphs with d^@?=3 (Aksenov, Mel@?nikov, and Steinberg, 1980). The first result in the positive direction of Havel@?s problem was made in 2003 by Borodin and Raspaud, who proved that every planar graph with d^@?>=4 and no 5-cycles is 3-colorable. Recently, Havel@?s problem was solved by Dvorak, Kral@? and Thomas in the positive, which means that there exists a constant d such that each planar graph with d^@?>=d is 3-colorable. (As far as we can judge, this d is very large.) We conjecture that the strongest possible version of Havel@?s problem (SVHP) is true: every planar graph with d^@?>=4 is 3-colorable. In this paper we prove that each planar graph with d^@?>=4 and without 5-cycles adjacent to triangles is 3-colorable. The readers are invited to prove a stronger theorem: every planar graph with d^@?>=4 and without 4-cycles adjacent to triangles is 3-colorable, which could possibly open way to proving SVHP.

Splits of circuits

This paper discusses an attempt at identifying a property of circuits in (nonplanar) graphs resembling the separation property of circuits in planar graphs derived from the Jordan Curve Theorem. If G is a graph and C is a circuit in G, we say that two circuits in G form a split of C if the symmetric difference of their edges sets is equal to the edge set of C, and if they are separated in G by the intersection of their vertex sets. Garcia Moreno and Jensen, A note on semiextensions of stable circuits, Discrete Math. 309 (2009) 4952-4954, asked whether such a split exists for any circuit C whenever G is 3-connected. We observe that if true, this implies a strong form of a version of the Cycle Double-Cover Conjecture suggested in the Ph.D. thesis of Luis Goddyn. The main result of the paper shows that the property holds for Hamilton circuits in cubic graphs.

Decompositions of graphs into trees, forests, and regular subgraphs

A conjecture by A. Hoffmann-Ostenhof suggests that any connected cubic graph G contains a spanning tree T for which every component of G - E ( T ) is a K 1 , a K 2 , or a cycle. We show that any cubic graph G contains a spanning forest F for which every component of G - E ( F ) is a K 2 or a cycle, and that any connected graph G ? K 1 with maximal degree at most 3 contains a spanning forest F without isolated vertices for which every component of G - E ( F ) is a K 1 , a K 2 or a cycle. We also prove a related statement about path-factorizations of graphs with maximal degree 3.

14. Chromatic Polynomials

If one were to delve into the fascinating combinatorial branch of mathematics that is graph theory, it would not take too long at all for one to encounter the infamous four colour theorem. It was proposed that for any planar graph, it would always be possible to colour any vertex in one of four colours such that no pair of vertices connected by a shared edge would be of the same colour. This theorem, formally proposed in 1852, had been left unproven for years, baffling the minds of great mathematicians for over a century, until it was rigorously proven in 1976 by Kenneth Appel and Wolfgang Haken, with the aid of computer calculation and algorithms which were obviously not to hand at the time [2]. In the decades that the proof was left undiscovered, many mathematicians have tried to tackle the problem with various approaches. One attempt proposed by George David Birkhoff in 1912, was the introduction of chromatic polynomials, a set of polynomials in variable λ which could be used to define the number of possible colourings of a given graph using λ colours. If Birkhoff could prove that every chromatic polynomial of a planar graph was strictly greater than zero in the case where λ = 4, then each planar graph would have at least one 4-colouring, which would prove the four colour theorem to be correct [1]. Sadly, Birkhoffs efforts were unfruitful for their initial goal. However, chromatic polynomials have been since further studied, with mathematicians achieving results that will be shown in this very paper, which have been helpful in finding the number of colourings for various graphs, compounds and augmentations.