Minor-equivalence for infinite graphs

Two graphs are minor-equivalent if each is isomorphic to a minor of the other. In this paper, we give structural characterizations of the minor equivalence classes of the infinite full grid GZ×L and of the infinite half-grid GZ×N. A corollary of these results states that every minor of GZ×z that has a minor isomorphic to GZ×N is minor-equivalent to one of GZ×Z or GZ×N. © 1999 Elsevier Science B.V. All rights reserved

A decomposition of locally finite graphs

We prove that every infinite, connected, locally finite graph G can be expressed as an edge-disjoint union of a leafless tree T, rooted at an arbitrarily chosen vertex of G, and a collection of finite graphs H1, H2, H3,...such that, for all i less than j, the vertices common to Hi and Hj lie in T, and no vertex of Hj lies on T between a vertex of Hi∩T and the root. © 1993

A note on intertwines of infinite graphs

We present a construction of two infinite graphs G1, G2 and of an infinite set F of graphs such that F is an antichain with respect to the minor relation and, for every graph G in F, both G1 and G2 are subgraphs of G but no graph obtained from G by deletion or contraction of an edge has both G1 and G2 as minors. These graphs show that the extension to infinite graphs of the intertwining conjecture of Lovász, Milgram, and Ungar fails. © 1993 Academic Press, Inc

Coloring graphs with crossings

We generalize the Five-Color Theorem by showing that it extends to graphs with two crossings. Furthermore, we show that if a graph has three crossings, but does not contain K6 as a subgraph, then it is also 5-colorable. We also consider the question of whether the result can be extended to graphs with more crossings. © 2008 Elsevier B.V. All rights reserved

On tree-partitions of graphs

A graph G admits a tree-partition of width k if its vertex set can be partitioned into sets of size at most k so that the graph obtained by identifying the vertices in each set of the partition, and then deleting loops and parallel edges, is a forest. In the paper, we characterize the classes of graphs (finite and infinite) of bounded tree-partition-width in terms of excluded topological minors

Unavoidable Parallel Minors of 4-Connected Graphs

A parallel minor is obtained from a graph by any sequence of edge contractions and parallel edge deletions. We prove that, for any positive integer k, every internally 4-connected graph of sufficiently high order contains a parallel minor isomorphic to a variation of K_{4,k} with a complete graph on the vertices of degree k, the k-partition triple fan with a complete graph on the vertices of degree k, the k-spoke double wheel, the k-spoke double wheel with axle, the (2k+1)-rung Mobius zigzag ladder, the (2k)-rung zigzag ladder, or K_k. We also find the unavoidable parallel minors of 1-, 2-, and 3-connected graphs.Comment: 12 pages, 3 figure

Unavoidable minors of graphs of large type

In this paper, we study one measure of complexity of a graph, namely its type. The type of a graph G is defined to be the minimum number n such that there is a sequence of graphs G = G0, G1,...,Gn, where Gi is obtained by contracting one edge in or deleting one edge from each block of G/_i, and where G is edgeless. We show that a 3-connected graph has large type if and only if it has a minor isomorphic to a large fan. Furthermore, we show that if a graph has large type, then it has a minor isomorphic to a large fan or to a large member of one of two specified families of graphs. © 2002 Elsevier Science B.V. All rights reserved