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On Mixed Cages of Girth 6

Gabriela Araujo-Pardo, Mirabel Mendoza-Cadena

TL;DR

The paper addresses the problem of determining or bounding the minimum order of mixed graphs with prescribed in/out degrees and undirected edges that attain a given girth, focusing on girth 6. It develops an infinite family of [z,r;6]-mixed graphs using the biaffine plane construction, establishing explicit upper bounds on the minimum order n[z,r;6] for certain (z,r). It also provides a general lower bound framework for [z,r;g]-mixed cages by combining circulant digraphs with Moore-style trees, and demonstrates a concrete [1,3;6]-mixed cage of 30 vertices derived from the Galois projective plane GF(4). The results advance understanding of mixed cages by delivering both new constructions with girth 6 and a holistic lower-bound perspective, with implications for planning near-optimal mixed cages in this regime.

Abstract

A [z,r;g]-mixed cage is a mixed graph of minimum order such that each vertex has z in-arcs, z out-arcs, r edges, and it has girth g. We present an infinite family of mixed graphs with girth 6. This construction also provides an upper bound on the minimum order of mixed cages of girth 6. Additionally,we introduce a lower bound on the minimum order for any mixed cage.

On Mixed Cages of Girth 6

TL;DR

The paper addresses the problem of determining or bounding the minimum order of mixed graphs with prescribed in/out degrees and undirected edges that attain a given girth, focusing on girth 6. It develops an infinite family of [z,r;6]-mixed graphs using the biaffine plane construction, establishing explicit upper bounds on the minimum order n[z,r;6] for certain (z,r). It also provides a general lower bound framework for [z,r;g]-mixed cages by combining circulant digraphs with Moore-style trees, and demonstrates a concrete [1,3;6]-mixed cage of 30 vertices derived from the Galois projective plane GF(4). The results advance understanding of mixed cages by delivering both new constructions with girth 6 and a holistic lower-bound perspective, with implications for planning near-optimal mixed cages in this regime.

Abstract

A [z,r;g]-mixed cage is a mixed graph of minimum order such that each vertex has z in-arcs, z out-arcs, r edges, and it has girth g. We present an infinite family of mixed graphs with girth 6. This construction also provides an upper bound on the minimum order of mixed cages of girth 6. Additionally,we introduce a lower bound on the minimum order for any mixed cage.
Paper Structure (13 sections, 7 theorems, 6 equations, 4 figures)

This paper contains 13 sections, 7 theorems, 6 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Previous work.
  3. Our results.
  4. Preliminaries
  5. The graph $\bm{G_{(2,q)}}$ of the projective plane on the Galois field.
  6. The bipartite graph $\bm{B_q}$.
  7. The circulant digraph $\overset{\bm{\rightarrow}}{\bm{C}}_{\bm{q}} \bm{{(i_1, \dots, i_k)}}$.
  8. The tree $\bm{\mathcal{T}_{r,g}}$ and lower Bounds for mixed cages.
  9. A family of mixed graphs with girth 6
  10. Lower Bound for mixed cages
  11. The [1,3;6]-mixed cage from Galois' Projective Plane
  12. Funding:
  13. Disclosure statement:

Key Result

Theorem 2

Let $n[1,r;g]$ be the order of a $[1,r;g]$-mixed cage. Then,

Figures (4)

  • Figure 1: $C_m$ for $m \in \mathbb{Z}_{11}$, $p = 5$. Node $[m,b]$ is written simply by $b$. Original nodes are drawn in orange and copy nodes in blue. Outside arcs have jumps of length 3 while inside arcs have jumps of length 5. One shortest path is $(0,2',5,7',10,10')$.
  • Figure 2: Construction for the $[1,3;6]$-mixed graph with 36 nodes.
  • Figure 3: Lower bound for a $[2,5;6]$-mixed cage with 11 + 66 - 6 = 69 nodes. Circulant graph has numbered vertices. The first level of vertices of the tree is shown in thick orange.
  • Figure 4: [1,3;6]-mixed cage from graph $G_{(2,4)}$. Erased edges and vertices are denoted by gray. Arcs are shown in blue. The new edge is shown in yellow. The rest of edges belong to the original graph $G_{(2,4)}$, where line $[m,b]$ colors points $(x_1,y_1)$ and $(x_2,y_2)$, and thus each point has two colors associated.

Theorems & Definitions (13)

  • Conjecture 1: Behzad-Chartrand-Wall, behzad1970minimal
  • Theorem 2: AHM lower bound,araujo2019mixed
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Corollary 5
  • Corollary 6
  • Theorem 7
  • proof
  • ...and 3 more