Suppose we have a graph \(G\) with vertex set \(V\) and edge set \(E\). We are going to show how the principle of inclusion and exclusion may be used to compute the number of ways to color a graph properly using colors from a set \(C\) of \(c\) colors. It is traditional to use the Greek letter \(γ\) (gamma) 2 to stand for the number of connected components of a graph in particular, \(γ(V, E)\) stands for the number of connected components of the graph with vertex set \(V\) and edge set \(E\). If we have a graph on the vertex set \(V\) with edge set \(E\) and another graph on the vertex set \(V\) with edge set \(E'\), then these two graphs could have different connected components. Notice that the connected components depend on the edge set of the graph. Thus we have a partition of the vertex set, and the blocks of the partition are the connected components of the graph. Since \(B_1 = B_2\), these two sets are the same block, and thus all blocks containing \(v\) are identical, so \(v\) is in only one block. (Relevant in Appendix C as well as this section.) Show that \(B_1 = B_2\). Then \(B_1\) is the set of all vertices connected by walks to some vertex \(v_1\) and \(B_2\) is the set of all vertices connected by walks to some vertex \(v_2\). To prove that we have defined a partition, suppose that vertex \(v\) is in the blocks \(B_1\) and \(B_2\). To have a partition, each vertex must be in one and only one block. Clearly, each vertex is in at least one block, because vertex \(v\) is connected to vertex \(v\) by the trivial walk consisting of the single vertex \(v\) and no edges. For each vertex \(v\) we put all vertices connected to it by a walk into a block together. Given a graph which might or might not be connected, we partition its vertices into blocks called connected components as follows. ![]() (Recall that a graph is connected if, for each pair of vertices, there is a walk between them.) Here, disconnected graphs will also be important to us. Here we are interested in a different, though related, problem: namely, in how many ways may we properly color a graph (regardless of whether it can be drawn in the plane or not) using \(k\) or fewer colors? When we studied trees, we restricted ourselves to connected graphs. You may have heard of the famous four color theorem of graph theory that says if a graph may be drawn in the plane so that no two edges cross (though they may touch at a vertex), then the graph has a proper coloring with four colors. A coloring is called proper if for each edge joining two distinct vertices 1, the two vertices it joins have different colors. A coloring of a graph by the elements of a set \(C\) (of colors) is an assignment of an element of \(C\) to each vertex of the graph that is, a function from the vertex set \(V\) of the graph to \(C\). ![]() ![]() We defined a graph to consist of set \(V\) of elements called vertices and a set \(E\) of elements called edges such that each edge joins two vertices. \)ĥ.2.4: The Chromatic Polynomial of a Graph
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