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An Inductive Proof of Whitney's Broken Circuit Theorem

Klaus Dohmen

TL;DR

This paper provides a self-contained inductive proof of Whitney's broken circuit theorem, giving a combinatorial interpretation of the coefficients $a_k(G)$ of the chromatic polynomial $P_G(\lambda)$ in terms of subsets avoiding broken circuits. The proof proceeds by induction on the number of edges and uses the deletion-contraction formula $P_G(\lambda)=P_{G-e}(\lambda)-P_{G|_e}(\lambda)$, together with a careful edge-contraction construction that preserves a fixed edge order. A bijective counting argument links $k$-subsets of $E(G)$ with no broken circuits to corresponding subsets in the deletion and contraction graphs, culminating in the stated coefficient interpretation. Compared with existing proofs via inclusion-exclusion or bijections like Blass–Sagan, this approach offers a simple, inductive route and clarifies the role of edge-ordering in the broken-circuit framework.

Abstract

We present a new proof of Whitney's broken circuit theorem based on induction on the number of edges and the deletion-contraction formula.

An Inductive Proof of Whitney's Broken Circuit Theorem

TL;DR

This paper provides a self-contained inductive proof of Whitney's broken circuit theorem, giving a combinatorial interpretation of the coefficients of the chromatic polynomial in terms of subsets avoiding broken circuits. The proof proceeds by induction on the number of edges and uses the deletion-contraction formula , together with a careful edge-contraction construction that preserves a fixed edge order. A bijective counting argument links -subsets of with no broken circuits to corresponding subsets in the deletion and contraction graphs, culminating in the stated coefficient interpretation. Compared with existing proofs via inclusion-exclusion or bijections like Blass–Sagan, this approach offers a simple, inductive route and clarifies the role of edge-ordering in the broken-circuit framework.

Abstract

We present a new proof of Whitney's broken circuit theorem based on induction on the number of edges and the deletion-contraction formula.

Paper Structure

This paper contains 3 sections, 6 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Whitney's broken circuit theorem

Key Result

Theorem 1

Let $G$ be a finite graph with a linear ordering relation on its edges. Then, for $k=0,\dots,n(G)$ the coefficient $a_k(G)$ equals the number of $k$-subsets of the edge-set of $G$ not including any broken circuit of $G$ as a subset.

Theorems & Definitions (12)

  • Theorem : Whitney, 1932
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 2 more