What is a Graph?

• A graph consists of two sets \(V\) and \(E\) where
• \[V = \{v_1, \ldots, v_n \}\]
• \[E = \{e_1, \ldots, e_m \}\]
• where each edge \(e_i\) consits of an unordered vertex pair \(e_i = \{ v_j, v_k \}\).
• \(j \neq k\) because we avoid self loops.
• For a directed graph we consider edges as ordered pairs.

Example

• Let \(V = \{ 1,2,3, 4,5 \}\) and
• \(E = \{ (1,2), (1,3), (2,3), (2,4) \}\).
• For an unordered graph, the order of the vertices in each edge does not matter.
• \(G = \{V, E\}\) is a graph.

Drawing a graph

Let us draw

Complete Graphs

• A complete graph means that every vertex is connected to every other vertex.
• In other words for all \(i,j\) we have \(e_k = \{v_i, v_j\} \in E\).
• We avoid self loops so \(i \neq j\).
• We denote them by \(K_n\).

Graph Vertex Coloring

• For a graph \(G = \{V,E \}\) a vertex coloring is a choice of colors
• \[ C = \{c_1, \ldots, c_p \}\]
• and an assignment of colors to vertices \(f : V \to C\)
• such that for any edge \(e = \{v_i, v_j \}\),
• \[ f(v_i) \neq f(v_j).\]
• In other words, for each edge we assign its constituent vertices with different colors.

Example

Question: Chromatic Number

• Given a graph \(G = \{ V, E \}\),
• what is the least number of colors to use to vertex color the graph?
• We denote the number by \(\chi(G)\).
• It is called the chromatic number for the graph.

Bounds on Chromatic Number

• Claim: \(\chi(G) \leq n\) where n is number of vertices of G.
• Proof:
• For complete graphs equality holds, \(\chi(G) =n\).
• Can we find a better bound?

contd..

• Let us look at the degrees at each vertex.
• Here degree of a vertex is the number of edges emanating from the vertex.
• Example:

contd..

• Suppose \(\Delta(G)\) is the largest degree of any vertex.
• \[ \chi(G) \leq \Delta(G) +1 .\]
• Can we show this?
• Answer: Use a greedy algorithm.

Application: Class Scheduling

• If there are \(n\) classes.
• Students signup for them without knowing the time schedules.
• How many distinct time slots to pick?
• Answer:

Answer:

• Pick vertices as each class.
• Join two vertices by an edge if there is a student taking both classes.
• Chromatic number gives the minimum number of distinct time slots.

LP Formulation

• Can you formulate an LP for the above problem?
• Say we have a graph \(G = (V,E)\).
• To start off we can pick \(n\) colors \(\{ c_1, \ldots, c_n \}\).
• Variables:
• \(y[i] = 1\) if we use color \(i\) and 0 otherwise.
• \(x[i,j] = 1\) if vertex \(i\) is assigned color \(j\), 0 otherwise

Chromatic LP

• Minimize $_{i} y[i] $.
• subject to
• \(\sum_k x[ik] = 1\) for every \(i\).
• \(x[ik] \leq y[k]\)
• \(x[ik] + x[jk] \leq 1\) for \((v_i, v_k) \in E\) and each \(k\).
• \(y[k] \leq y[k-1]\).
• \(x[ik], y[k] \in \{0,1 \}\).

Edge Coloring

• Question: How many colors do we need to color the edges of a graph such that no two edges of the same color connect at a vertex?
• Example:

Bounds on Edge Coloring

• We need at least \(\Delta(G)\) colors.
• Infact this is a very good estimate.
• Vizing in 1964 proved that the number of colors needed is either
• \(\Delta(G)\) or
• $(G) +1 $.

Edge Coloring Example

• Schedule matches in a tournament.
• Suppose we have 9 teams and we have a graph saying which teams must play each other.
• How many rounds of play needed to achieve this?

Example:

LP Formulation

• It is similar to vertex LP except this time we do for edges.