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Circuit partitions and signed interlacement in 4-regular graphs

Lorenzo Traldi

TL;DR

This workextends the theory of circuit partitions in 4-regular graphs by introducing a real-valued counterpart to the standard GF(2) interlacement framework. It proves that there exists an integer matrix $M_{\mathbb{R}}(C,P)$ (reducing to the existing $M(C,P)$ mod $2$) whose row space equals the cycle space of the touch-graph $Tch(P)$ over $\mathbb{R}$ and whose $\mathbb{R}$-nullity equals $|P|-c(F)$, thereby generalizing the circuit-nullity relation beyond orientability. It also presents a practical standard form, $M_{\mathbb{R}}^{0}(C,P)$, with detailed structure and naturality properties under κ-transformations and transpositions, and demonstrates the oriented case where $M_{\mathbb{R}}^{0}(C,P)$ connects to skew-symmetric interlacement matrices. The results yield a determinant-based method to count Euler systems compatible with a given edge orientation and deepen algebraic characterizations of circle-graph-like structures via touch-graphs and interlacement.

Abstract

Let $F$ be a 4-regular graph. Each circuit partition $P$ of $F$ has a corresponding touch-graph $Tch(P)$; the circuits in $P$ correspond to vertices of $Tch(P)$, and the vertices of $F$ correspond to edges of $Tch(P)$. We discuss the connection between modified versions of the interlacement matrix of an Euler system of $F$ and the cycle space of $Tch(P)$, over $GF(2)$ and $\mathbb{R}$.

Circuit partitions and signed interlacement in 4-regular graphs

TL;DR

This workextends the theory of circuit partitions in 4-regular graphs by introducing a real-valued counterpart to the standard GF(2) interlacement framework. It proves that there exists an integer matrix (reducing to the existing mod ) whose row space equals the cycle space of the touch-graph over and whose -nullity equals , thereby generalizing the circuit-nullity relation beyond orientability. It also presents a practical standard form, , with detailed structure and naturality properties under κ-transformations and transpositions, and demonstrates the oriented case where connects to skew-symmetric interlacement matrices. The results yield a determinant-based method to count Euler systems compatible with a given edge orientation and deepen algebraic characterizations of circle-graph-like structures via touch-graphs and interlacement.

Abstract

Let be a 4-regular graph. Each circuit partition of has a corresponding touch-graph ; the circuits in correspond to vertices of , and the vertices of correspond to edges of . We discuss the connection between modified versions of the interlacement matrix of an Euler system of and the cycle space of , over and .

Paper Structure

This paper contains 8 sections, 17 theorems, 41 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Statement of the main theorem
  3. Proof of the main theorem
  4. A standard form for $M_{\mathbb{R}}(C,P)$
  5. Four examples
  6. The effect of a $\kappa$-transformation
  7. The effect of a transposition
  8. The oriented case

Key Result

Theorem 2

Suppose $C$ is an Euler system of a 4-regular graph $F$, and $P$ is a circuit partition of $F$. Let $\mathcal{I}(C,P)$ be the symmetric $GF(2)$-matrix obtained from $\mathcal{I}(C)$ by making these two kinds of changes. Then the $GF(2)$-nullity of $\mathcal{I}(C,P)$ is $\left\vert P\right\vert -c(F)$, where $\left\vert P\right\vert$ is the number of circuits in $P$ and $c(F)$ is the number of con

Figures (4)

  • Figure 1: An Euler circuit $C$, a 3-element circuit partition $P_1$ and a 2-element circuit partition $P_2$. To follow a circuit, maintain the same plain/dashed line status when traversing a vertex.
  • Figure 2: Touch-graphs from Figure \ref{['circfig']}.
  • Figure 3: $F$ is on the left, and $Tch(P)$ is on the right.
  • Figure 4: The directed touch-graph from the second example.

Theorems & Definitions (34)

  • Definition 1
  • Theorem 2
  • Definition 3
  • Definition 4
  • Theorem 5
  • Lemma 6
  • proof
  • Theorem 7
  • Theorem 8
  • Proposition 9
  • ...and 24 more