Complete Minors and Independence Number
نویسنده
چکیده
Let G be a graph with n vertices and independence number α. Hadwiger’s conjecture implies that G contains a clique minor of order at least n/α. In 1982, Duchet and Meyniel proved that this bound holds within a factor 2. Our main result gives the first improvement on their bound by an absolute constant factor. We show that G contains a clique minor of order larger than .504n/α. We also prove related results giving lower bounds on the order of the largest clique minor.
منابع مشابه
Independence number and clique minors
Since χ(G) · α(G) ≥ |V (G)|, Hadwiger’s Conjecture implies that any graph G has the complete graph Kdn α e as a minor, where n is the number of vertices of G and α is the maximum number of independent vertices in G. Motivated by this fact, it is shown that any graph on n vertices with independence number α ≥ 3 has the complete graph Kd n 2α−2 e as a minor. This improves the well-known theorem o...
متن کاملToo Many Minor Order Obstructions ( For Parameterized Lower
We study the growth rate on the number obstructions (forbidden minors) for families of graphs that are based on parameterized graph problems. Our main result shows that if the de ning graph problem is NP-complete then the growth rate on the number of obstructions must be super-polynomial or else the polynomial-time hierarchy must collapse to P 3 . We illustrate the rapid growth rate of paramete...
متن کاملLarge minors in graphs with given independence number
Aweakening of Hadwiger’s conjecture states that every n-vertex graphwith independence number α has a clique minor of size at least n α . Extending ideas of Fox (2010) [6], we prove that such a graph has a clique minor with at least n (2−c)α vertices where c > 1/19.2. © 2011 Elsevier B.V. All rights reserved.
متن کاملLarge Minors in Graphs with a given Stability Number
A weakening of Hadwiger’s Conjecture states that every n-vertex graph with independence number α has a clique minor of size at least n α . Extending ideas of Fox [6], we prove that such a graph has a clique minor with at least n (2−c)α vertices where c > 1/19.2.
متن کاملIndependence systems with continuous cardinality of bases
The maximum cardinality basis problem in a genoid (i.e. an independence system), although NP-complete in general, is solvable in a polynomial time in the following cases: matchings of a graph [l], intersection of two matroi’ds [2,4] and independent set of vertices in claw-free graphs [S, 71. These three genoi’ds satisfy the continuous cardinal&y of bases for all their minors. The aim of this pa...
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:
- SIAM J. Discrete Math.
دوره 24 شماره
صفحات -
تاریخ انتشار 2010