Deletion–contraction formula

As some illustrative examples, the number of spanning trees t(G) satisfies DC.[3] Note below we use a slightly different definition of arboricity that refers to spanning trees (as opposed to forests):

Proof for 'arboricity' ${\displaystyle t(G)}$

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t(G \ e) denotes the number of spanning trees not including e, whereas t(G / e) the number including e. To see the second, if T is an st of G then contracting e produces another st of G / e. Conversely, if we have an st T of G / e, then expanding the edge e gives two disconnected trees; adding e connects the two and gives an st of G.

The chromatic polynomial ${\displaystyle \chi _{k}(G)}$ counting the number of k-colorings of G is DC, but for a slightly modified formula (which can be made equivalent):[1] ${\displaystyle \chi (G)=\chi (G\backslash e)-\chi (G/e)}$

Proof for chromatic polynomial ${\displaystyle \chi _{k}(G)}$

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If e = uv, then a k-coloring of G is the same as a k-coloring of G \ e where u and v have different colors. There are ${\displaystyle \chi (G\backslash e)}$ total G \ e colorings. We need now subtract the ones where u and v are colored similarly. But such colorings correspond to the k-colorings of ${\displaystyle \chi (G/e)}$ where u and v are merged.

This above property can be used to show that the chromatic polynomial ${\displaystyle \chi _{k}(G)}$ is indeed a polynomial in k. We can do this via induction on the number of edges and noting that in the base case where there are no edges, there are ${\displaystyle k^{|V(G)|}}$ possible colorings (which is a polynomial in k).

• Inclusion–exclusion principle
• Tutte polynomial
• chromatic polynomial
• Nowhere-zero flow