The Extremal Function for Complete Minors

Interval minors of complete bipartite graphs

Interval minors of bipartite graphs were recently introduced by Jacob Fox in the study of Stanley-Wilf limits. We investigate the maximum number of edges in Kr,s-interval minor free bipartite graphs. We determine exact values when r = 2 and describe the extremal graphs. For r = 3, lower and upper bounds are given and the structure of K3,s-interval minor free graphs is studied.

Cliques in Graphs Excluding a Complete Graph 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...

Extremal Positive Solutions For The Distributed Order Fractional Hybrid Differential Equations

In this article, we prove the existence of extremal positive solution for the distributed order fractional hybrid differential equation$$int_{0}^{1}b(q)D^{q}[frac{x(t)}{f(t,x(t))}]dq=g(t,x(t)),$$using a fixed point theorem in the Banach algebras. This proof is given in two cases of the continuous and discontinuous function $g$, under the generalized Lipschitz and Caratheodory conditions.

Graphs Without Large Complete Minors Are Quasi-Random

We answer a question of S ´ os by showing that, if a graph G of order n and density p has no complete minor larger than would be found in a random graph G(n, p), then G is quasi-random, provided either p > 0.45631. .. or κ(G) n(log log log n)/(log log n), where 0.45631. .. is an explicit constant. The results proved can also be used to fill the gaps in an argument of Thomason, describing the ex...

Extremal Functions for Graph Linkages and Rooted Minors

Bounds on Minors of Binary Matrices

We prove an upper bound on sums of squares of minors of {+1, −1}-matrices. The bound is sharp for Hadamard matrices, a result due to de Launey and Levin [‘(1, −1)-matrices with near-extremal properties’, SIAM J. Discrete Math. 23 (2009), 1422–1440], but our proof is simpler. We give several corollaries relevant to minors of Hadamard matrices. 2010 Mathematics subject classification: primary 05B...