On semi-transitive orientability of circulant graphs

A graph $G = (V, E)$ is said to be word-representable if a word $w$ can be formed using the letters of the alphabet $V$ such that for every pair of vertices $x$ and $y$, $xy \in E$ if and only if $x$ and $y$ alternate in $w$. A \textit{semi-transitive} orientation is an acyclic directed graph where for any directed path $v_0 \rightarrow v_1 \rightarrow \ldots \rightarrow v_m$, $m \ge 2$ either there is no arc between $v_0$ and $v_m$ or for all $1 \le i < j \le m$ there is an arc between $v_i$ and $v_j$. An undirected graph is semi-transitive if it admits a semi-transitive orientation. For given positive integers $n, a_1, a_2, \ldots, a_k$, we consider the undirected circulant graph with set of vertices ${0, 1, 2, \ldots, n-1}$ and the set of edges${ij ~ | ~ (i - j) \pmod n$ or $(j-i) \pmod n$ are in ${a_1, a_2, \ldots, a_k}}$, where $ 0 < a_1 < a_2 < \ldots < a_k < (n+1)/2$. Recently, Kitaev and Pyatkin have shown that every $4$-regular circulant graph is semi-transitive. Further, they have posed an open problem regarding the semi-transitive orientability of circulant graphs for which the elements of the set ${a_1, a_2, \ldots, a_k}$ are consecutive positive integers. In this paper, we solve the problem mentioned above. In addition, we show that under certain assumptions, some $k(\ge5)$-regular circulant graphs are semi-transitive, and some are not. Moreover, since a semi-transitive orientation is a characterisation of word-representability, we give some upper bound for the representation number of certain $k$-regular circulant graphs.

💡 Research Summary

The paper investigates the semi‑transitive orientability of circulant graphs C(n; a₁,…,a_k) and its close connection to word‑representability. A semi‑transitive orientation is an acyclic digraph in which any directed path of length at least two either has no shortcut edge from its first to its last vertex or, if such an edge exists, all intermediate vertices are pairwise adjacent. Because a graph is word‑representable exactly when it admits a semi‑transitive orientation, the authors study both properties simultaneously.

The main focus is on circulant graphs whose distance set R consists of consecutive integers. The authors first recall known results: every 4‑regular circulant graph is semi‑transitive, and a circulant graph is bipartite (hence transitive) precisely when all distances are odd and n is even.

Negative result (Theorem 2.3).
For parameters satisfying 2 < (n+1)/5 ≤ t ≤ n−1/4, the graph C(n; t, t+1,…, 2t) is not semi‑transitive. The proof extracts an induced subgraph on six vertices {0, t−1, t, 2t−1, 2t+1, n−t} and shows it is isomorphic to W₅, a known non‑semi‑transitive graph. Hence any larger graph containing this subgraph cannot be semi‑transitive. This settles the open problem of Kitaev and Pyatkin in the negative for a wide range of t.

Positive results.

Theorem 2.4 proves that if the smallest distance a₁ satisfies a₁ ≥ ⌈n/4⌉, then the orientation that directs every edge from the smaller to the larger vertex is semi‑transitive. The argument uses a simple recurrence: along any directed path v₀→v₁→…→v_m (m ≥ 3) each step increases the vertex label by at least n/4, forcing v_m ≥ v₀+3·(n+1)/4, which contradicts the existence of a shortcut edge v₀→v_m that would require v_m ≤ v₀+3·n/4.

Theorem 2.5 extends this to the extreme case where the distance set is the whole interval {t, t+1,…,⌊n/2⌋}. Again the “small‑to‑large’’ orientation works: any directed path satisfies v₀ + t·ℓ ≤ v_ℓ ≤ v₀ + ℓ·(n−t). If a shortcut existed, the intermediate vertices would all be adjacent, forming a clique, which contradicts the definition of a shortcut. Consequently, all such circulants are semi‑transitive.

These two theorems together give a complete picture for consecutive distance sets: when the interval is short (roughly up to twice the smallest distance) the graph may be non‑semi‑transitive; when the interval starts sufficiently far from zero or covers the whole half‑range, the graph is always semi‑transitive. This generalises the earlier result that every 4‑regular circulant graph (which corresponds to distance set {1, 2}) is semi‑transitive, and shows that many k‑regular circulants with k ≥ 5 are also semi‑transitive under natural numeric conditions.

Word‑representability and representation numbers.

The authors then turn to explicit word constructions.

Theorem 3.1 shows that C(n; 1, 2,…, k) is 2‑word‑representable. They define a morphism h(i)=i(i−k)ⁿ and apply it to the linear order 0 1 … n−1, obtaining a 2‑uniform word w. By analysing the subword between the two occurrences of each letter i, they verify that exactly the neighbours of i appear there, establishing the required alternation property.

For 3‑regular circulants of the form C(2n; a, n) with gcd(a, 2n)=1, Theorem 3.2 constructs a 3‑uniform word using the morphism h(i)=i(i−a)²ⁿ(i+n)²ⁿ. The resulting word represents the graph, proving that the representation number R(G) ≤ 3.

Theorem 3.3 proves that the same graphs are not 2‑word‑representable when n > 2. Assuming a 2‑uniform representation, one can select a vertex whose two copies have no other letter between them; Observation 1.11 forces the three neighbours of that vertex to be exactly the letters between the copies. Because these three neighbours form an independent set, a careful case analysis leads to a contradiction, showing that a 2‑uniform word cannot exist. Hence R(G)=3 for these graphs.

The paper concludes with several open questions, such as whether every k‑regular circulant graph has representation number at most k, and how to pinpoint the exact threshold for semi‑transitivity when the distance set is not consecutive.

Overall, the work provides a thorough classification of semi‑transitive orientability for circulant graphs with consecutive distance sets, extends known results to higher regularities, and supplies concrete word constructions that bound the representation numbers of important families of circulant graphs. It deepens the understanding of the interplay between graph orientations and combinatorial word representations.