3. Consider the following graph Find its chromatic number k x(G) and find a k-critical subgraph:
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First, we need to determine the chromatic number of the graph. The chromatic number is the minimum number of colors needed to color the vertices of the graph such that no two adjacent vertices have the same color. Looking at the graph, we can see that there are Show more…
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A graph G is said to be k-critical if χ(G) = k and the deletion of any vertex yields a graph of smaller chromatic number. Suppose G is k-critical. Show δ(G) ≥ k-1, (the smallest vertex-degree is bigger than or equal to k-1)
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Show that if $G$ is a chromatically $k$ -critical graph, then the degree of every vertex of $G$ is at least $k-1$ .
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