In this video we prove by induction that every graph has chromatic number at most one more than the maximum degree. Odd cycles and complete graphs are examples for which the chromatic number meets this upper bound exactly. For other graphs, Brook's Theo...

From Sarada Herke

In this video we begin by showing that the chromatic number of a tree is 2. Yet, if the chromatic number of a graph is 2, this does not imply that the graph is a tree. We then prove that the chromatic number of a graph is 2 if and only if the graph is b...

From Sarada Herke

In this video, we begin with a visualisation of an edge contraction and discuss the fact that an edge contraction may be thought of as resulting in a multigraph or simple graph, depending on the application. We then state the definition a contraction of ...

From Sarada Herke

We have seen in a previous video that K5 and K3,3 are non-planar. In this video we define an elementary subdivision of a graph, as well as a subdivision of a graph. We then discuss the fact that if a graph G contains a subgraph which is a subdivision of...

From Sarada Herke

In this video we formally prove that the complete graph on 5 vertices is non-planar. Then we prove that a planar graph with no triangles has at most 2n-4 edges, where n is the number of vertices. Using this fact, we formally prove that the complete bipa...

From Sarada Herke

An isomorphism from a graph G to a graph H is a bijection from the vertex set of G to the vertex set of H such that adjacency and non-adjacency are preserved. However, finding such a mapping is also equivalent to find a permutation matrix P such that A =...

From Sarada Herke

In a connected plane graph with n vertices, m edges and r regions, Euler's Formula says that n-m+r=2. In this video we try out a few examples and then prove this fact by induction. We discuss a generalization to disconnected plane graphs as well as what ...

From Sarada Herke

This is the talk I gave for "The Laborastory" series about science stories and great science heroes.
The hero I chose to talk about is Sal Khan, the creator of Khan Academy and the educator that has revolutionised on-line education.
--A talk by Dr. Sarad...

From Sarada Herke

In video 02: Definition of a Graph, we defined a (simple) graph as a set of vertices together with a set of edges where the edges are 2-subsets of the vertex set. Notice that this definition does not allow for multiple edges or loops. In general on this...

From Sarada Herke

Recall that a block in a graph is a maximal nonseparable subgraph, and the centre of a graph is the set of all vertices whose eccentricity is equal to the radius (minimum eccentricity). In this video we walk through a proof that the centre of every conne...

From Sarada Herke

In this video we look at two terms which are related to the idea of cut-vertices in a graph. Firstly, an edge is a bridge if its removal from a graph creates more connected components than were previously there. We have seen the topic of bridge edges in...

From Sarada Herke

Notice that the complete graph on n vertices has no cut-vertices, whereas the path on n vertices (where n is at least 3) has n-2 cut-vertices. Can you ever have a connected graph with more than n-2 cut-vertices? The answer is no. We prove that in any n...

From Sarada Herke

Eccentricity, radius and diameter are terms that are used often in graph theory. They are related to the concept of the distance between vertices. The distance between a pair of vertices is the length of a shortest path between them. We begin by review...

From Sarada Herke

Here we describe the difference between two similar sounding words in mathematics: maximum and maximal. We use concepts in graph theory to highlight the difference. In particular, we define an independent set in a graph and a component in a graph and lo...

From Sarada Herke

In this video I describe the motivation behind Gray codes, which are a specific way of listing the binary sequences of length n. We also discuss a Graph-Theoretic way of viewing Gray codes.
-- Bits of Discrete Math by Dr. Sarada Herke.
Related videos:
...

From Sarada Herke

In this video I define the complement of a graph and what makes a graph self-complementary. I show some examples, for orders 4 and 5 and discuss a necessary condition on the order of a graph for it to be self-complementary. Finally I give a brief descri...

From Sarada Herke

A quick guide to some important mathematical notation, especially for discrete math, combinatorics and graph theory. I use small examples to review notation used in counting, set theory, summation and products, quantifiers and logic.
-- by Dr. Sarada H...

From Sarada Herke

It is not surprising that a tree of order k is a subgraph of a complete graph of order at least k. Here I'll explain the result that shows for every tree T of order k, any graph with minimum degree at least k-1 will contain a subgraph isomorphic to T. T...

From Sarada Herke

In this video we describe how to work with graphs in Sage, which is a very useful free mathematical software based on the Python programming language. In particular, in this video we show how to check if a pair of graphs are isomorphic using the Sage sof...

From Sarada Herke

Here I provide two examples of determining when two graphs are isomorphic. If they are isomorphic, I give an isomorphism; if they are not, I describe a property that I show occurs in only one of the two graphs.
Here is a related video in which I show h...

From Sarada Herke

In this video I provide the definition of what it means for two graphs to be isomorphic. I illustrate this with two isomorphic graphs by giving an isomorphism between them, and conclude by discussing what it means for a mapping to be a bijection.
An intr...

From Sarada Herke

Here we provide solutions to a basic problem set in Graph Theory. This part 2 of 2 answers the following:
3) For k = 0,1 and 2, characterize the k-regular graphs.
4) Prove that in a bipartite graph with partite sets X and Y, the sum of the degrees of the ...

From Sarada Herke

We give the definition of a connected graph and give examples of connected and disconnected graphs. We also discuss the concepts of the neighbourhood of a vertex and the degree of a vertex. This allows us to define a regular graph, and we give some exam...

From Sarada Herke

This video describes some important families of graph in Graph Theory, including Complete Graphs, Bipartite Graphs, Paths and Cycles.
--An introduction to Graph Theory by Dr. Sarada Herke.
Links to the related videos:
https://www.youtube.com/watch?v=S1Z...

From Sarada Herke

In this video we formally define what a graph is in Graph Theory and explain the concept with an example. In this introductory video, no previous knowledge of Graph Theory will be assumed.
--An introduction to Graph Theory by Dr. Sarada Herke.
This vide...

From Sarada Herke