Interlacement in 4-regular graphs: a new approach using nonsymmetric matrices

Let F be a 4-regular graph with an Euler system C. We introduce a simple way to modify the interlacement matrix of C so that every circuit partition P of F has an associated modified interlacement matrix M(C,P ). If C and C′ are Euler systems of F then M(C,C′) and M(C′, C) are inverses, and for any circuit partition P , M(C′, P ) = M(C′, C) · M(C,P ). This machinery allows for short proofs of several results regarding the linear algebra of interlacement. 1. Interlacement and local complements A graph G = (V (G), E(G)) is given by a finite set V (G) of vertices, and a finite set E(G) of edges. In a looped simple graph each edge is incident on one or two vertices, and different edges have different vertex-incidences; an edge incident on only one vertex is a loop. A simple graph is a looped simple graph with no loop. In general, a graph may have parallel edges (distinct edges with the same vertex-incidences). Edge-vertex incidences generate an equivalence relation on E(G) ∪ V (G); the equivalence classes are the connected components of G, and the number of connected components is denoted c(G). Two vertices incident on a non-loop edge are neighbors, and if v ∈ V (G) then N(v) = {neighbors of v} is the open neighborhood of v. Each edge consists of two distinct half-edges, and the edge has two distinct directions given by designating one half-edge as initial and the other as terminal. Each half-edge is incident on a vertex; if the edge is not a loop then the half-edges are incident on different vertices. The number of halfedges incident on a vertex v is the degree of v, and a d-regular graph is one whose vertices all have degree d. In a directed graph each vertex has an indegree and an outdegree; a d-in, d-out digraph is one whose vertices all have indegree d and outdegree d. A circuit in a graph is a sequence v1, h1, h ′ 1, v2, ..., vk, hk, h ′ k, vk+1 = v1 such that for each i, hi+1 and h ′ i are half-edges incident on vi+1, and hi and h ′ i are the half-edges of an edge ei; ei 6= ej when i 6= j. A directed circuit in a directed graph is a circuit in which hi is the initial half-edge of ei, for every i. An Euler circuit is a Received by the editors July 14, 2012, and in revised form February 4, 2014. 2010 Mathematics Subject Classification. 05C31.