Total Chromatic Numbers of Cₙ(1,3) Graphs

• The paper establishes that for Cₙ(1,3), 5 colors suffice for all n ≥ 7 except for n ∈ {7,8,12,13,17}, which require 6 colors.
• Explicit cyclic-block constructions are used to demonstrate valid total colorings, ensuring vertices and edges receive distinct colors.
• Obstruction arguments in exceptional cases reveal key combinatorial limitations, reinforcing the bounds of the Total Coloring Conjecture.

A total coloring of a graph is an assignment of colors to all vertices and edges such that no two adjacent vertices, no two adjacent edges, and no edge together with either of its incident vertices share the same color. The total chromatic number, denoted $\chi''(G)$, is the smallest number of colors required for such a coloring of $G$. The circulant graph $C_{n}(1, 3)$ is defined on $n$ vertices with edges joining each $v_i$ to $v_{i+1}$ and $v_{i+3}$ (indices mod $n$), making it $4$-regular. Determining $\chi''(C_{n}(1, 3))$ has been a central problem for four-regular circulant graphs. Recent advances have resolved this problem precisely for all $n \ge 7$, showing that $\chi''(C_{n}(1, 3)) = 6$ for five exceptional $n$ and $5$ otherwise, as established in Su–Tong–Yang (Su et al., 24 Dec 2025) and in the infinite subfamilies by Navaneeth–Geetha–Somasundaram–Fu (Navaneeth et al., 2021).

1. Formal Definition and Background

The total coloring problem generalizes the classical vertex and edge coloring concepts by requiring conflict-free assignments across both sets. In this setting:

• A total $k$-coloring assigns $k$ colors to $V(G) \cup E(G)$ such that:
  • adjacent vertices differ,
  • adjacent edges differ,
  • each edge and its endpoints differ.
• The total chromatic number $\chi''(G)$ is the minimal such $k$.
• For any graph $G$ with maximum degree $\Delta(G)$, the known bounds are $\Delta+1 \leq \chi''(G) \leq \Delta+2$.

The Total Coloring Conjecture (TCC) by Behzad and Vizing posits that $\chi''(G) \leq \Delta(G) + 2$ for all graphs.

The circulant $C_{n}(1, 3)$, with vertex set $\{v_0,v_1,\ldots,v_{n-1}\}$ and edges $E_1: v_i v_{i+1}$ and $E_2: v_i v_{i+3}$, is $4$-regular for $n \ge 7$, so $5 \leq \chi''(C_{n}(1, 3)) \leq 6$.

2. Main Results: Exact Values of $\chi''(C_{n}(1, 3))$

The precise determination of the total chromatic number of $C_{n}(1, 3)$ relies on both explicit coloring constructions and obstruction arguments:

• For all $n \ge 7$ except $n \in \{7,8,12,13,17\}$, one has

$\chi''(C_{n}(1, 3)) = 5.$

• For $n \in \{7,8,12,13,17\}$, one has

$\chi''(C_{n}(1, 3)) = 6,$

which conforms with the Kostochka bound and matches previously computed values for small instances.

This result is established for all $n \ge 7$ by Su–Tong–Yang (Su et al., 24 Dec 2025) and for infinite subfamilies (such as $n=3p$ for $p$ odd, and $n=9p$ for $p \ge 1$) by Navaneeth–Geetha–Somasundaram–Fu (Navaneeth et al., 2021).

Summary Table

n	$\chi''(C_{n}(1, 3))$
7, 8, 12, 13, 17	6
all other $n \ge 7$	5

3. Constructions: Achieving the Lower Bound

For all admissible $n$ except the five exceptions, explicit periodic colorings establish $\chi''=5$.

General Case ($n = 5p + 9q$):

• Vertex colors: $(24351)^p \cdot (212534121)^q$ cyclically.
• Edges of type $E_1$: $(12123)^p \cdot (453453453)^q$ cyclically.
• Edges of type $E_2$: $(45534)^p \cdot (121212534)^q$ cyclically.

Crucially, these sequences are selected such that local neighborhoods do not repeat color among a vertex and its incident edges. For specific $n$ not covered by periodic templates (namely $n=11,16,21,26,31,22$), separate explicitly enumerated colorings are provided (see (Su et al., 24 Dec 2025), Lemma 4).

Editor's term: The “cyclic-block” template is the central tool in these constructions. It leverages static color assignment schemes extended around the cycle and checked locally for valid total coloring properties.

4. Exceptional Values: Obstruction Arguments

For $n \in \{7,8,12,13,17\}$, lower bounds are enforced by combinatorial obstructions:

• $n=7$: Independence number is $2$, but Lemma 3.2 (Chetwynd–Hilton) forces five odd-sized color classes, contradicting the possibility of assembling a total coloring with just five colors.
• $n=8$: Isomorphic to $K_{4,4}$, whose total chromatic number is known to be $6$.
• $n=12$: Previously computed to require $6$ colors.
• $n=13,17$: Detailed distance and color class enumeration produces an inevitable conflict at some vertex, prohibiting a total $5$-coloring.

These are rigorously justified either by classical results or structured case analyses ((Su et al., 24 Dec 2025), Section 5).

5. Infinite Subfamilies and Theoretical Significance

Prior work had characterized the total chromatic numbers for four-regular circulant graphs in various cases:

• $C_{n}(2k,3)$ for specific divisibility conditions [Junior & Sasaki 2020]
• $C_{n}(1,k)$ for $k \equiv 1$ or $2 \pmod{3}$ with $\gcd(n,k) = 1$ [Khennoufa & Togni 2008]

Navaneeth–Geetha–Somasundaram–Fu (Navaneeth et al., 2021) completed the classification for the $k \equiv 0 \pmod{3}$ cases, notably showing that:

• If $n=3p$, $p$ odd, or $n=9p$, one always has $\chi''(C_{n}(1,3)) = 5$.

This result demonstrates that in every infinite subfamily with $n$ divisible by $3$, $C_{n}(1,3)$ is Type I ($\chi'' = \Delta+1$), confirming the TCC in these instances and extending the known typology for such graphs.

6. Implications for Graph Coloring Theory

The resolution of $\chi''(C_{n}(1,3))$ for all $n$ delivers a complete answer for this family, marking a conclusive advance in total graph coloring of circulant graphs. The result highlights:

• The rarity of Type II behavior (i.e., requiring $\Delta+2$ colors), restricted to five small orders.
• The sufficiency of uniform cyclic constructions for almost all $n$.
• The persistent value of structural combinatorial lemmas in precluding small total colorings for exceptional graph sizes.

A plausible implication is that similar periodic or cyclic coloring templates may generalize to broader families of regular circulants, subject to combinatorial constraints visible in small graphs. This suggests ongoing relevance for symmetry-based coloring constructions in regular graphs.