Constructing dense graphs with sublinear Hadwiger number

Mader asked to explicitly construct dense graphs for which the size of the largest clique minor is sublinear in the number of vertices. Such graphs exist as a random graph almost surely has this property. This question and variants were popularized by Thomason over several articles. We answer these questions by showing how to explicitly construct such graphs using blow-ups of small graphs with this property. This leads to the study of a fractional variant of the clique minor number, which may be of independent interest.Comment: 10 page

Disproof of the List Hadwiger Conjecture

The List Hadwiger Conjecture asserts that every $K_t$-minor-free graph is $t$-choosable. We disprove this conjecture by constructing a $K_{3t+2}$-minor-free graph that is not $4t$-choosable for every integer $t\geq 1$

Hadwiger Number and the Cartesian Product Of Graphs

The Hadwiger number mr(G) of a graph G is the largest integer n for which the complete graph K_n on n vertices is a minor of G. Hadwiger conjectured that for every graph G, mr(G) >= chi(G), where chi(G) is the chromatic number of G. In this paper, we study the Hadwiger number of the Cartesian product G [] H of graphs. As the main result of this paper, we prove that mr(G_1 [] G_2) >= h\sqrt{l}(1 - o(1)) for any two graphs G_1 and G_2 with mr(G_1) = h and mr(G_2) = l. We show that the above lower bound is asymptotically best possible. This asymptotically settles a question of Z. Miller (1978). As consequences of our main result, we show the following: 1. Let G be a connected graph. Let the (unique) prime factorization of G be given by G_1 [] G_2 [] ... [] G_k. Then G satisfies Hadwiger's conjecture if k >= 2.log(log(chi(G))) + c', where c' is a constant. This improves the 2.log(chi(G))+3 bound of Chandran and Sivadasan. 2. Let G_1 and G_2 be two graphs such that chi(G_1) >= chi(G_2) >= c.log^{1.5}(chi(G_1)), where c is a constant. Then G_1 [] G_2 satisfies Hadwiger's conjecture. 3. Hadwiger's conjecture is true for G^d (Cartesian product of G taken d times) for every graph G and every d >= 2. This settles a question by Chandran and Sivadasan (They had shown that the Hadiwger's conjecture is true for G^d if d >= 3.)Comment: 10 pages, 2 figures, major update: lower and upper bound proofs have been revised. The bounds are now asymptotically tigh

Average degree conditions forcing a minor

Mader first proved that high average degree forces a given graph as a minor. Often motivated by Hadwiger's Conjecture, much research has focused on the average degree required to force a complete graph as a minor. Subsequently, various authors have consider the average degree required to force an arbitrary graph $H$ as a minor. Here, we strengthen (under certain conditions) a recent result by Reed and Wood, giving better bounds on the average degree required to force an $H$-minor when $H$ is a sparse graph with many high degree vertices. This solves an open problem of Reed and Wood, and also generalises (to within a constant factor) known results when $H$ is an unbalanced complete bipartite graph

A relative of Hadwiger's conjecture

Hadwiger's conjecture asserts that if a simple graph $G$ has no $K_{t+1}$ minor, then its vertex set $V(G)$ can be partitioned into $t$ stable sets. This is still open, but we prove under the same hypotheses that $V(G)$ can be partitioned into $t$ sets $X_1,\ldots, X_t$, such that for $1\le i\le t$, the subgraph induced on $X_i$ has maximum degree at most a function of $t$. This is sharp, in that the conclusion becomes false if we ask for a partition into $t-1$ sets with the same property.Comment: 6 page