[Paper Review] On a certain representation of the chromatic polynomial

Yu. V. Matiyasevich|ArXiv.org|Mar 6, 2009

Graph theory and applications3 references3 citations

TL;DR

This paper presents a novel representation of the chromatic polynomial C(G,k) using flow polynomials F(H,k) over edge subgraphs H of a graph G, establishing a deep algebraic connection between chromatic and flow polynomials. The key result is an exact formula expressing C(G,k) as a weighted sum over subgraphs H, with coefficients involving F(H,k) and powers of (1−k), revealing a duality in planar graphs via geometric duality and the chromatic polynomial of the dual graph.

ABSTRACT

The representation is essentially the same as that given by J.P.Nagle in J. Comb. Theory (B), 1971, 10:1, 42--59. The distinction is in the definition of the weighting function via the number of flows. This new definition allows one to deduce a number of corollaries, in particular, the following. A) The chromatic polynomial of a connected planar graph G can be uniquely determined from its combinatory dual graph G^* (although the graph G itself isn't, in general, determined uniquely by G^*). B) If a planar graph G is different from the full graph K_3 and has exactly one (up to renaming of colors) proper coloring of vertices in three colors, then the graph G^* dual to graph G is also vertex colorable in three colors.

Motivation & Objective

• To establish a new algebraic representation of the chromatic polynomial C(G,k) using flow polynomials F(H,k) over edge subgraphs H of G.
• To clarify the relationship between the weight function w(H,k) introduced in prior work and standard graph-theoretic invariants like flow polynomials.
• To demonstrate how this representation simplifies computation of chromatic polynomials, especially in planar graphs.
• To reveal a duality between chromatic polynomials of planar graphs and their geometric duals via the flow polynomial.

Proposed method

• The paper derives a new formula for the chromatic polynomial C(G,k) as a sum over all edge subgraphs H ≤ G, weighted by F(H,k)/(1−k)^m(H), scaled by (k−1)^m(G)/k^{m(G)−n(G)}.
• It proves that the weight function w(H,k) from earlier work satisfies w(H,k) = k^{m(H)−n(H)} F(H,k), thereby linking it to the flow polynomial.
• For planar graphs, the paper uses geometric duality to show that F(H,k) = C(H*,k)/k, where H* is the dual of H, leading to a dual representation of C(G,k).
• It establishes a congruence modulo (k−1)^2 for graphs with m(G) > 1, showing C(G,k) ≡ (−1)^m C(G*,k) mod (k−1)^2.
• The proof relies on counting balanced flows and proper colorings via algebraic structures over the ring R_k, using the correspondence between colorings and flows on dual graphs.
• It uses Euler’s formula and edge-duality to map subgraphs H ≤ G to contractions L ≼ G* in the dual graph, enabling the transformation between the two forms of the formula.

Experimental results

Research questions

• RQ1 How can the chromatic polynomial be expressed in terms of flow polynomials of subgraphs?
• RQ2 What is the precise algebraic relationship between the weight function w(H,k) and the flow polynomial F(H,k)?
• RQ3 How does the chromatic polynomial of a planar graph relate to the chromatic polynomial of its dual graph?
• RQ4 Can the chromatic polynomial be simplified modulo (k−1)^2 using this representation?
• RQ5 What structural properties of graphs (e.g., presence of isthmuses) affect the behavior of the weight function w(H,k)?

Key findings

• The chromatic polynomial C(G,k) is exactly represented as C(G,k) = (k−1)^m(G)/k^{m(G)−n(G)} × ∑_{H≤G} F(H,k)/(1−k)^m(H), establishing a direct link between chromatic and flow polynomials.
• The weight function w(H,k) from prior work satisfies w(H,k) = k^{m(H)−n(H)} F(H,k), which explains its known properties such as vanishing on graphs with isthmuses.
• For planar graphs, the formula transforms into C(G,k) = (k−1)^m(G)/k^{m(G)−n(G)+1} × ∑_{H≤G} C(H*,k)/(1−k)^m(H), revealing a duality between G and its dual G*.
• When m(G) > 1, C(G,k) ≡ (−1)^m C(G*,k) mod (k−1)^2, showing a strong congruence between the chromatic polynomials of dual graphs.
• The result implies that if a planar graph G has exactly one proper 3-coloring (up to renaming), then its dual G* is also 3-colorable, as shown by substituting k=3.
• The correspondence between proper colorings of G* and balanced flows on G is bijective up to a factor of k, and this bijection is constructed via dual edge orientations and spanning trees.