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On the Word-Representability of 5-Regular Circulant Graphs

Suchanda Roy, Ramesh Hariharasubramanian

TL;DR

This paper extends the theory of word-representable graphs to 5-regular circulant graphs by leveraging number theory, group theory, graph colorings, morphisms, and factorization. It systematically analyzes $C_{2n}(a,b,n)$ by parity and gcd, establishes 3-colorability in many cases, and constructs explicit 5-uniform representations via morphisms with several subfamilies achieving $R(G)\le5$. It also shows how Cartesian factorization into smaller circulants preserves word-representability and yields bounds on the representation number, broadening known results for lower-regular circulants. The work concludes with open problems and a conjecture that all 5-regular circulants are word-representable, outlining pathways to tackle higher-regularity cases.

Abstract

A graph $G = (V, E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that, for any two distinct vertices $x, y \in V$, $xy \in E$ if and only if $x$ and $y$ alternate in $w$. Two letters $x$ and $y$ are said to alternate in $w$ if, after removing all other letters from $w$, the resulting word is of the form $xyxy\dots$ or $yxyx\dots$ (of even or odd length). For a given set $R = \{r_1, r_2, \dots, r_k\}$ of jump elements, an undirected circulant graph $C_n(R)$ on $n$ vertices has vertex set $\{0, 1, \dots, n-1\}$ and edge set $ E = \left\{ \{i,j\} \;\middle|\; |i - j| \bmod n \in \{r_1, r_2, \dots, r_k\} \right\}, $ where $0 < r_1 < r_2 < \dots < r_k < \frac{n}{2}$. Recently, Kitaev and Pyatkin proved that every 4-regular circulant graph is word-representable. Srinivasan and Hariharasubramanian further investigated circulant graphs and obtained bounds on the representation number for $k$-regular circulant graphs with $2 \le k \le 4$. In addition to these positive results, their work also presents examples of non-word-representable circulant graphs. In this work, we study word-representability and the representation number of 5-regular circulant graphs via techniques from elementary number theory and group theory, as well as graph coloring, graph factorization and morphisms.

On the Word-Representability of 5-Regular Circulant Graphs

TL;DR

This paper extends the theory of word-representable graphs to 5-regular circulant graphs by leveraging number theory, group theory, graph colorings, morphisms, and factorization. It systematically analyzes by parity and gcd, establishes 3-colorability in many cases, and constructs explicit 5-uniform representations via morphisms with several subfamilies achieving . It also shows how Cartesian factorization into smaller circulants preserves word-representability and yields bounds on the representation number, broadening known results for lower-regular circulants. The work concludes with open problems and a conjecture that all 5-regular circulants are word-representable, outlining pathways to tackle higher-regularity cases.

Abstract

A graph is word-representable if there exists a word over the alphabet such that, for any two distinct vertices , if and only if and alternate in . Two letters and are said to alternate in if, after removing all other letters from , the resulting word is of the form or (of even or odd length). For a given set of jump elements, an undirected circulant graph on vertices has vertex set and edge set where . Recently, Kitaev and Pyatkin proved that every 4-regular circulant graph is word-representable. Srinivasan and Hariharasubramanian further investigated circulant graphs and obtained bounds on the representation number for -regular circulant graphs with . In addition to these positive results, their work also presents examples of non-word-representable circulant graphs. In this work, we study word-representability and the representation number of 5-regular circulant graphs via techniques from elementary number theory and group theory, as well as graph coloring, graph factorization and morphisms.

Paper Structure

This paper contains 13 sections, 41 theorems, 40 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Word-Representable Graphs
  4. Circulant Graph
  5. Word-representability of 5-regular Circulant Graph
  6. Word-Representability and Colorability of 5-Regular Circulant Graphs
  7. Word-Representability and the Representation Number of 5-Regular Circulant Graph Using Morphisms
  8. Word-representability of 5-Regular Circulant Graph Through Factorization
  9. Conclusion and Future Work
  10. Appendix
  11. Proof of Theorem 25.
  12. Proof of Theorem 26.
  13. Proof of Theorem 27.

Key Result

theorem 1

A linear congruence of the form $b x \equiv a \pmod{n}$ has a unique solution modulo $n$ if and only if $\gcd(b, n) = 1$.

Figures (6)

  • Figure 1: A semi-transitive orientation of $K_4$
  • Figure 2: Schematic structure of the circulant graph with partitions $X,Y$ and $Z$.
  • Figure 3: Color assignments for the divisions of $X, Y, Z$
  • Figure 4: Color assignments for the divisions of $X, Y, Z$ except the vertices $n, 2n-x$ and $2n-1$.
  • Figure 4: A semi-transitive orientation of $K_6$.
  • ...and 1 more figures

Theorems & Definitions (77)

  • theorem 1
  • definition 1
  • proposition 1
  • definition 2
  • proposition 2
  • definition 3: kitaev2015words
  • proposition 3: kitaev2015words
  • definition 4: kitaev2015words
  • definition 5: kitaev2015words
  • definition 6: kitaev2015words
  • ...and 67 more