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What is and is not inside a Cayley graph?

Kolja Knauer, Alvaro Soto Gomez

TL;DR

The paper investigates which graphs can occur as subgraphs of minimal Cayley graphs, addressing Babai's questions on chromatic number and structure. It introduces faithful no-lonely-color colorings of Cayley digraphs as a necessary condition for embedding, and develops algorithms to test these conditions. The authors construct a cubic graph of girth $5$ that cannot be embedded into any minimal Cayley graph and prove that Generalized Petersen Graphs $G(n,k)$ with $ abla(n,k)=1$ embed as induced subgraphs of minimal Cayley graphs (via $C_n times C_r$). They also present computational results clarifying the embedding status of specific graphs and pose open questions on the general existence of excluded subgraphs and the universality of Petersen graphs in this context.

Abstract

In this note we show that there is a cubic graph of girth $5$ that is not a subgraph of any minimal Cayley graph. On the other hand, we show that any Generalized Petersen Graph $G(n,k)$ with $\gcd(n,k)=1$ is an induced subgraph of a minimal Cayley graph. These results give insights into two comments of László Babai in [L. Babai, \emph{Automorphism groups, isomorphism, reconstruction}. Graham, R. L. (ed.) et al., Handbook of combinatorics. Vol. 1-2, 1994].

What is and is not inside a Cayley graph?

TL;DR

The paper investigates which graphs can occur as subgraphs of minimal Cayley graphs, addressing Babai's questions on chromatic number and structure. It introduces faithful no-lonely-color colorings of Cayley digraphs as a necessary condition for embedding, and develops algorithms to test these conditions. The authors construct a cubic graph of girth that cannot be embedded into any minimal Cayley graph and prove that Generalized Petersen Graphs with embed as induced subgraphs of minimal Cayley graphs (via ). They also present computational results clarifying the embedding status of specific graphs and pose open questions on the general existence of excluded subgraphs and the universality of Petersen graphs in this context.

Abstract

In this note we show that there is a cubic graph of girth that is not a subgraph of any minimal Cayley graph. On the other hand, we show that any Generalized Petersen Graph with is an induced subgraph of a minimal Cayley graph. These results give insights into two comments of László Babai in [L. Babai, \emph{Automorphism groups, isomorphism, reconstruction}. Graham, R. L. (ed.) et al., Handbook of combinatorics. Vol. 1-2, 1994].

Paper Structure

This paper contains 7 sections, 3 theorems, 1 equation, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. No-lonely-color colorings of digraphs
  3. An algorithm
  4. Two cubic graphs of girth 5
  5. Generalized Petersen Graphs
  6. Questions
  7. Acknowledgments:

Key Result

Lemma 2.3

If $D=(V,A)$ is subdigraph of a minimal Cayley digraph $\overrightarrow{\textrm{Cay}}(\Gamma,S)$, then assigning via $f:A\to S$ to each $a\in A$ the corresponding element of $S$ is a faithful no-lonely-color.

Figures (2)

  • Figure 1: Left: The triplex graph is not subgraph of a minimal Cayley graph. Right: An orientation of the twinplex graph with a coloring from our algorithm.
  • Figure 2: Left: the filled vertices induce $G(7,2)$ in $\textrm{Cay}(C_7\rtimes C_3,\{a,b\})$. Right: The Petersen plane, where white dots are Cayley graphs, Grey dots are induced subgraph of a minimal Cayley graph of $C_n\rtimes C_r$ by Theorem \ref{['thm:Petersen']}, and black dots have been obtained by computer.

Theorems & Definitions (9)

  • Definition 2.1
  • Lemma 2.3
  • proof
  • Definition 2.2
  • Lemma 2.5
  • Theorem 5.1
  • proof
  • Conjecture 6.1
  • Conjecture 6.2