Every Graph of Sufficiently Large Average Degree Contains a C4-Free Subgraph of Large Average Degree
نویسندگان
چکیده
We prove that for every k there exists d = d(k) such that every graph of average degree at least d contains a subgraph of average degree at least k and girth at least six. This settles a special case of a conjecture of Thomassen.
منابع مشابه
Semester Report
This is best possible in the sense that we cannot additionally require the entire bipartite subgraph of G between S and T to have large minimum degree. Applying the result of Mader that every graph of sufficiently large average degree contains a subdivision of a given graph H to the graph G[S] obtained by the above theorem, it immediately follows that every highly connected graph G contains a “...
متن کاملLarge Topological Cliques in Graphs Without a 4-Cycle
Mader asked whether every C4-free graph G contains a subdivision of a complete graph whose order is at least linear in the average degree of G. We show that there is a subdivision of a complete graph whose order is almost linear. More generally, we prove that every Ks,t-free graph of average degree r contains a subdivision of a complete graph of order r 1 2 + 1 2(s−1).
متن کاملLarge Cliques in C4-Free Graphs
A graph is called C4-free if it contains no cycle of length four as an induced subgraph. We prove that if a C4-free graph has n vertices and at least c1n 2 edges then it has a complete subgraph of c2n vertices, where c2 depends only on c1. We also give estimates on c2 and show that a similar result does not hold for H-free graphs—unless H is an induced subgraph of C4. The best value of c2 is de...
متن کاملA proof of Mader's conjecture on large clique subdivisions in C4-free graphs
Given any integers s, t ≥ 2, we show there exists some c = c(s, t) > 0 such that any Ks,t-free graph with average degree d contains a subdivision of a clique with at least cd 1 2 s s−1 vertices. In particular, when s = 2 this resolves in a strong sense the conjecture of Mader in 1999 that every C4-free graph has a subdivision of a clique with order linear in the average degree of the original g...
متن کاملSubdivisions of a large clique in C6-free graphs
Mader conjectured that every C4-free graph has a subdivision of a clique of order linear in its average degree. We show that every C6-free graph has such a subdivision of a large clique. We also prove the dense case of Mader’s conjecture in a stronger sense, i.e. for every c, there is a c′ such that every C4-free graph with average degree cn 1/2 has a subdivision of a clique K` with ` = bc′n1/2...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Combinatorica
دوره 24 شماره
صفحات -
تاریخ انتشار 2004