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A short class with Sarah and her students. The other ways are under More.
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Today, we will explore edge colouring in graphs. Can anyone tell me what edge colouring means?
I believe it's when we assign colours to edges in a graph so that no two edges that share a vertex have the same colour.
Exactly! Edge colouring helps us to organize connections in a graph. Now, does anyone know if there are limits to how many edges we can colour with the same colour?
Isn't it related to the number of vertices?
Yes! We can only colour a certain number of edges with the same colour depending on whether we have an even or odd number of vertices. Remember: for even n, we can't exceed n/2 + 1 edges.
And what about odd n?
Great question! For odd n, the limit changes. We'll dive deeper into that. Let's summarize: edge colouring requires careful consideration of vertex parity.
Now, let's discuss the edge chromatic number for complete graphs. Can anyone explain what this is?
It’s the minimum number of colours needed to colour the edges of a graph.
Exactly! For complete graphs, if you have an even number of vertices, how many colours do we need at least?
At least n - 1 colours, right?
Right again! And if n is odd, how many colours will that require, based on our earlier discussions?
We would need at least n colours since we can’t use a single colour for more than (n-1)/2 edges.
Fantastic! Today, we’ve learned how to determine the number of colours needed based on the properties of the graph. Keep these principles in mind for our exercises.
Next, let's look at some constructive examples of edge colouring. Suppose we have a complete graph with 8 vertices. How would we schedule matches using edge colouring?
We could assign matches for teams so that no team plays more than once a day.
Exactly! By assigning colours strategically, we can ensure that teams don't overlap in games. On the first day, we can colour edges connecting team 1, 2, 3, and 4 to team 8.
And then shift them for the next day, right?
Yes! Rotation helps us manage scheduling effectively. Can someone summarize how this method satisfies our edge colouring requirements?
By rotating, we can cover all edges without conflict, and it also meets the total colour number for the complete graph.
Exactly! Keep practicing these techniques, as they'll be vital for your understanding of graph theory.
Now let's address edge colouring for graphs where n is odd, say 7 vertices. Who can tell me how we might approach this?
We can add a dummy vertex to make the count even, like adding an 8th team.
Yes! By adding the dummy vertex, we can apply the same colours as if we had an even graph. What happens after we apply this method?
We can remove the dummy vertex and adjust the colouring for the odd count.
Excellent! This strategy optimizes our colouring while adhering to the edge chromatic number rules. Let’s summarize: adding a vertex can help manage odd numbers effectively.
The limits on edge colouring depend on whether the number of vertices is even or odd.
A single colour cannot be used to colour more than a defined number of edges, based on the parity.
This defines how many colours are minimally needed to avoid conflicting edges.
In a graph so neat and wide, edges must not overlap with pride. Colour them well, side to side, or conflicts they shall surely provide.
Imagine scheduling a tournament with teams where no one can play against two others at once. Each evening, teams gather, wearing different colours, happily playing one game each without clashes.
EVE - Even vertices limit edges to n/2 + 1, Odd vertices need n colours.
CLEAN can remind us about Colouring Limits: C = Colours, L = Limit, E = Edges, A = Assignments, N = Nodes.
Term
What is edge colouring?
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Term
What defines the edge chromatic number?
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The process of assigning colours to edges of a graph so that no two edges sharing the same vertex have the same colour.
The minimum number of colours required to colour the edges of a graph.
A graph in which every pair of distinct vertices is connected by a unique edge.
The fundamental units of graphs, represented as points or nodes.
This section discusses the principles of edge colouring in graphs, specifically focusing on the conditions when a single colour can be used for a set of edges based on the parity of the number of vertices.
In this section, we explore edge colouring in graphs with n vertices. By analyzing even and odd scenarios, we demonstrate that for even n, at most n/2 edges can be coloured with a single colour, while for odd n, the restrictions change, affecting the colour assignment. The section examines the implications on the edge chromatic number for complete graphs, and provides constructive examples to illustrate these principles.
In graph theory, edge colouring involves assigning colours to edges so that no two edges sharing a common vertex receive the same colour. The section primarily covers the following key points:
Single Colour Limitations: The section asserts that in a graph with n vertices, the use of a single colour to colour edges is limited based on the parity of n:
Edge Chromatic Number: The edge chromatic number is examined for complete graphs. The section discusses two cases:
Constructive Examples: Examples are provided to illustrate how the described edge colouring applies to complete graphs in real scenarios, such as scheduling using round robin methods.