Deletion–contraction formula

In graph theory, a deletion-contraction formula / recursion is any formula of the following recursive form:

${\displaystyle f(G)=f(G\setminus e)+f(G/e).}$

Here G is a graph, f is a function on graphs, e is any edge of G, G \ e denotes edge deletion, and G / e denotes contraction. Tutte refers to such a function as a W-function. ^[1] The formula is sometimes referred to as the fundamental reduction theorem. ^[2] In this article we abbreviate to DC.

R. M. Foster had already observed that the chromatic polynomial is one such function, and Tutte began to discover more, including a function f = t(G) counting the number of spanning trees of a graph (also see Kirchhoff's theorem). It was later found that the flow polynomial is yet another; and soon Tutte discovered an entire class of functions called Tutte polynomials (originally referred to as dichromates) that satisfy DC. ^[1]

Examples

Spanning trees

The number of spanning trees ${\displaystyle t(G)}$ satisfies DC. ^[3]

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

Laplacian characteristic polynomials

By Kirchhoff's theorem, the number of spanning trees in a graph is counted by a cofactor of the Laplacian matrix. However, the Laplacian characteristic polynomial does not satisfy DC. By studying Laplacians with vertex weights, one can find a deletion-contraction relation between the scaled vertex-weighted Laplacian characteristic polynomials. ^[4]

Chromatic polynomials

The chromatic polynomial ${\displaystyle \chi _{G}(k)}$ counting the number of k-colorings of G does not satisfy DC, but a slightly modified formula (which can be made equivalent): ^[1]

${\displaystyle \chi _{G}(k)=\chi _{G-e}(k)-\chi _{G/e}(k).}$

Proof. 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 ${\displaystyle \chi _{G\setminus e}(k)}$ 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 ${\displaystyle \chi _{G/e}(k)}$ where u and v are merged.

This above property can be used to show that the chromatic polynomial ${\displaystyle \chi _{G}(k)}$ 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).

See also

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

Citations

1. Tutte, W.T. (January 2004). "Graph-polynomials". Advances in Applied Mathematics. 32 (1–2): 5–9. doi: 10.1016/S0196-8858(03)00041-1 .
2. Dong, Koh & Teo (2005)
3. "Deletion-contraction and chromatic polynomials" (PDF). Archived from the original (PDF) on 4 May 2019.
4. Aliniaeifard, Farid; Wang, Victor; Stephanie van Willigenburg (2021). "Deletion-contraction for a unified Laplacian and applications". arXiv: 2110.13949 [math.CO].

Works cited

• Dong, F M; Koh, K M; Teo, K L (June 2005). Chromatic Polynomials and Chromaticity of Graphs. World Scientific. doi:10.1142/5814. ISBN   978-981-256-317-0.