Maximizing the Number of Independent Sets of a Fixed Size
نویسندگان
چکیده
Let it(G) be the number of independent sets of size t in a graph G. Engbers and Galvin asked how large it(G) could be in graphs with minimum degree at least δ. They further conjectured that when n ≥ 2δ and t ≥ 3, it(G) is maximized by the complete bipartite graph Kδ,n−δ. This conjecture has drawn the attention of many researchers recently. In this short note, we prove this conjecture.
منابع مشابه
INDEPENDENT SETS OF SOME GRAPHS ASSOCIATED TO COMMUTATIVE RINGS
Let $G=(V,E)$ be a simple graph. A set $Ssubseteq V$ isindependent set of $G$, if no two vertices of $S$ are adjacent.The independence number $alpha(G)$ is the size of a maximumindependent set in the graph. In this paper we study and characterize the independent sets ofthe zero-divisor graph $Gamma(R)$ and ideal-based zero-divisor graph $Gamma_I(R)$of a commutative ring $R$.
متن کاملMaximizing the Number of Independent Sets of Fixed Size in Connected Graphs with Given Independence Number
The Turán connected graph TCn,α is obtained from α cliques of size b αc or d αe by joining all cliques by an edge to one central vertex in one of the larger cliques. The graph TCn,α was shown recently by Bruyère and Mélot to maximise the number of independent sets among connected graphs of order n and independence number α. We prove a generalisation of this result by showing that TCn,α in fact ...
متن کاملCounting the number of independent sets in chordal graphs
We study some counting and enumeration problems for chordal graphs, especially concerning independent sets. We first provide the following efficient algorithms for a chordal graph: (1) a linear-time algorithm for counting the number of independent sets; (2) a linear-time algorithm for counting the number of maximum independent sets; (3) a polynomial-time algorithm for counting the number of ind...
متن کاملLinear-Time Counting Algorithms for Independent Sets in Chordal Graphs
We study some counting and enumeration problems for chordal graphs, especially concerning independent sets. We first provide the following efficient algorithms for a chordal graph: (1) a linear-time algorithm for counting the number of independent sets; (2) a linear-time algorithm for counting the number of maximum independent sets; (3) a polynomial-time algorithm for counting the number of ind...
متن کاملMatchings and independent sets of a fixed size in regular graphs
We use an entropy based method to study two graph maximization problems. We upper bound the number of matchings of fixed size ` in a d-regular graph on N vertices. For 2` N bounded away from 0 and 1, the logarithm of the bound we obtain agrees in its leading term with the logarithm of the number of matchings of size ` in the graph consisting of N 2d disjoint copies of Kd,d. This provides asympt...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Combinatorics, Probability & Computing
دوره 24 شماره
صفحات -
تاریخ انتشار 2015