We show that a graph of girth greater than 6 log k + 3 and minimum degree at least 3 has a minor of minimum degree greater than k. This is best possible up to a factor of at most 9/4. As a corollary, every graph of girth at least 6 log r + 3 log log r + c and minimum degree at least 3 has a Kr minor.

This paper considers the following question: What is the maximum number of k-cliques in an n-vertex graph with no Kt-minor? This question generalises the extremal function for Kt-minors, which corresponds to the k = 2 case. The exact answer is given for t 6 9 and all values of k. We also determine the maximum total number of cliques in an n-vertex graph with no Kt-minor for t 6 9. Several obser...

It is well known that in finite graphs, large complete minors/topological minors can be forced by assuming a large average degree. Our aim is to extend this fact to infinite graphs. For this, we generalize the notion of the relative end degree, which had been previously introduced by the first author for locally finite graphs, and show that large minimum relative degree at the ends and large mi...

We give an algorithm that, for a fixed graph H and integer k, decides whether an n-vertex Hminor-free graph G contains a path of length k in 2 √ k) · n steps. Our approach builds on a combination of Demaine-Hajiaghayi’s bounds on the size of an excluded grid in such graphs with a novel combinatorial result on certain branch decompositions of H-minor-free graphs. This result is used to bound the...

Let IF be field, and let A and B be n X n matrices with elements in IF. Suppose that A is completely reducible and that B is symmetric. If the principal minors of A determined by the 1and 2-circuits of the graph of B and by the chordless circuits of the graph of A are equal to the corresponding principal minors of B, then A is diagonally similar to B; and conversely.

A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to each other color class. The b-chromatic number of G is the maximum integer χb(G) for which G has a b-coloring with χb(G) colors. A graph G is b-continuous if G has a b-coloring with k colors, for every integer k in the interval [χ(G), χb(G)]. It is known that not all graphs...

We study the maximum flow problem in directed H-minorfree graphs where H can be drawn in the plane with one crossing. If a structural decomposition of the graph as a clique-sum of planar graphs and graphs of constant complexity is given, we show that a maximum flow can be computed in O(n logn) time. In particular, maximum flows in directed K3,3-minor-free graphs and directed K5-minor-free graph...

We give the first polynomial time approximation scheme for the subset Traveling Salesperson Problem (subset TSP) in H-minor-free graphs. Our main technical contribution is a polynomial time algorithm that, given an edge-weighted H-minor-free graph G and a set of k terminals T , finds a subgraph of G with weight at most OH(poly( 1 ) log k) times the weight of the minimum Steiner tree for T that ...