Encoding and Enumerating Acyclic Orientations of Graphs

TL;DR

The paper tackles the enumeration and encoding of acyclic orientations across graphs, with a focus on complete multipartite graphs. It introduces a length-$n$ encoding for unlabelled complete multipartite graphs, proves a linear-time construction from the encoding, and derives closed formulas for the numbers of non-isomorphic orientations and those containing a directed spanning tree; it further extends to labelled vertices using generating functions and Stirling-number frameworks. A universal permutation-code approach is developed for general graphs, enabling efficient encoding/decoding of acyclic orientations and yielding sharp upper and lower bounds $2^{n-1}\le A(G)\le n!$, along with Turán-type and chromatic-number–dependent bounds. Overall, the work provides concrete encodings, counting formulas, and algorithmic procedures that facilitate enumeration and analysis of acyclic orientations in multipartite and general graphs.

Abstract

In this work we study the acyclic orientations of graphs. We obtain an encoding of the acyclic orientations of the complete $p$-partite graph with size of its parts $n_1,n_2,\ldots,n_p$ via a vector with $p$ symbols and length $n=n_1+n_2+\ldots+n_p$ when the parts are fixed but not the vertices in each part. We also give a recursive way to construct all acyclic orientations of a complete multipartite graph, this construction can be done by computer easily in order $\mathcal{O}(n)$. Furthermore, we obtain a closed formula for non-isomorphic acyclic orientations of both the complete multipartite graphs and the complete multipartite graphs with a directed spanning tree. Moreover, we obtain a closed formula for the number of acyclic orientations of a complete multipartite graph $K_{n_1,\ldots,n_p}$ with labelled vertices. Finally, we obtain a way encode all acyclic orientations of an arbitrary graph as a permutation code. Using the codification mentioned above we obtain sharp upper and lower bounds of the number of acyclic orientations of a graph.

Figures (4)

• Figure 1: Two sources in consecutive orientations $A\to B$ and $B\to C$ for visualizing the argument in Claim 2.
• Figure 2: An acyclic orientation of $K_{2,2,1}$ (5-wheel graph) with consecutive sources: $x_1, y_1, x_2, y_2$ respectively
• Figure 3: An acyclic orientation of $K_{2,3}$ with sequences of sources $(2,3,1,4,5)$ or $(2,3,1,5,4)$, and code $0.1.0.1.1$
• Figure 4: Acyclic orientation with permutation (sources) code $(1,2,3,4,5)$ and permutation (sinks) code $(6,5,2,4,3,1)$

Theorems & Definitions (33)

• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Corollary 2.3
• proof
• Proposition 2.4
• proof
• Theorem 2.5
• proof