On \u3cem\u3eK\u3c/em\u3e\u3cem\u3e\u3csub\u3es,t\u3c/sub\u3e\u3c/em\u3e-minors in Graphs with Given Average Degree

Let D(H) be the minimum d such that every graph G with average degree d has an H-minor. Myers and Thomason found good bounds on D(H) for almost all graphs H and proved that for \u27balanced\u27 H random graphs provide extremal examples and determine the extremal function. Examples of \u27unbalanced graphs\u27 are complete bipartite graphs Ks,t for a fixed s and large t. Myers proved upper bounds on D(Ks,t ) and made a conjecture on the order of magnitude of D(Ks,t ) for a fixed s and t → ∞. He also found exact values for D(K2,t) for an infinite series of t. In this paper, we confirm the conjecture of Myers and find asymptotically (in s) exact bounds on D(Ks,t ) for a fixed s and large t

A proof of Mader's conjecture on large clique subdivisions in C4C_4-free graphs

Given any integers $s,t\geq 2$, we show there exists some $c=c(s,t)>0$ such that any $K_{s,t}$-free graph with average degree $d$ contains a subdivision of a clique with at least $cd^{\frac{1}{2}\frac{s}{s-1}}$ vertices. In particular, when $s=2$ this resolves in a strong sense the conjecture of Mader in 1999 that every $C_4$-free graph has a subdivision of a clique with order linear in the average degree of the original graph. In general, the widely conjectured asymptotic behaviour of the extremal density of $K_{s,t}$-free graphs suggests our result is tight up to the constant $c(s,t)$.Comment: 25 pages, 1 figur

Subdivisions in a bipartite graph

Given a bipartite graph G with m and n vertices, respectively,in its vertices classes, and given two integers s, t such that 2 ≤ s ≤ t, 0 ≤ m−s ≤ n−t, and m+n ≤ 2s+t−1, we prove that if G has at least mn−(2(m−s)+n−t) edges then it contains a subdivision of the complete bipartite $K_(s,t)$ with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn − (2(m − s) + n − t + 1) edges for this topological Turan type problem.Peer Reviewe