Prove that the independence number of a tree of order n is at least ⌈n / 2].

Name: Problem 33, Chapter 12: Introductory Combinatorics
Uploaded: 2020-08-16T13:56:13-08:00
Duration: 13 min 31 s
Channel: Chris Trentman
Description: Prove that the independence number of a tree of order n is at least ⌈n / 2].

Prove that the independence number of a tree of order n is...

Key Concepts

Tree

A tree is a connected, acyclic graph and serves as a fundamental structure in graph theory. Its acyclic property guarantees a unique path between any two vertices, which leads to many important properties, such as being inherently bipartite. Trees are widely used in various mathematical and practical applications, including network design and data structures.

Independence Number

The independence number of a graph is defined as the size of the largest set of vertices with no two vertices adjacent. This parameter is fundamental in many graph-theoretic problems, as it quantifies the largest group of mutually non-interacting elements. Studying the independence number leads to insights in areas such as scheduling, resource allocation, and combinatorial optimization.

Bipartite Graph

A bipartite graph is one whose vertices can be partitioned into two disjoint sets, such that no two vertices within the same set are adjacent. Since every tree is bipartite, one of the partitions forms an independent set. This property directly helps in establishing bounds on the independence number of trees by ensuring that at least one of the partitions has a size of at least half the total number of vertices.

Lower Bound Techniques in Graphs

Lower bound techniques in graph theory involve proving that a graph parameter, such as the independence number, must be at least as large as a certain value. In the case of trees, by leveraging the bipartite nature and the partitioning of vertices, one can show that one of the partitions (and thus an independent set) contains at least half of all vertices, thereby providing a lower bound of the ceiling of n/2 for the independence number.

Transcript

00:01 You're asked to prove that solon's algorithm produces a man -bang tree and a connected undirected weighted graph.

00:10 So we're essentially proving that solon's algorithm does what it says it does.

00:19 So to do this, first, let g be a connected, undirected, weighted graph, and now let s be the tree that's produced by solon's algorithm.

01:17 Let e1 through en minus 1 be its edges in the order chosen.

01:45 And we have the edges that were chosen in the same stage.

01:48 We really don't care about how these are ordered.

01:50 It's just arbitrary.

01:52 What matters is that edges that were in one stage come before edges in a stage that came later.

02:00 And now, let tb, a minimum spanning tree that contains all the edges, e1 through ek, where k lies between zero and n -m -m -m -m -spanning tree.

02:47 Minus 1.

03:04 Now i'll assume that k is less than n minus 1 and we'll derive a contradiction.

03:29 Now let s prime be the forest resulting from solon's algorithm for s at the stage before the edge ek plus 1 is added.

04:33 And now let's see be the component in s prime that results from the addition of ek plus 1.

05:15 And now let ek plus 1 be labeled u, v as an edge.

05:29 With u, of course, is going to line the component c.

05:36 I guess you should say instead of resulting from this is responsible for, and we have that v of course is not going to lie in c by solon's algorithm.

06:06 Now, let p be the unique path from u to v in t.

06:12 And this exists.

06:13 Because t is a tree...

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