Orientations of graphs omitting non-edge-critical directed graphs
Hannah Sheats
TL;DR
This work advances the understanding of how many edge-orientations of a graph avoid a fixed directed subgraph $\vec{H}$, by establishing a general framework that bounds $D(n,\vec{H})$ via the Turán-exponent of the underlying graph $H$ and the decomposition family $\mathcal{M}'(\vec{H})$. The authors prove two main theorems that yield near-Turán-type extremal behavior for large $n$ when $\mathcal{M}'(\vec{H})$ is finite and contains suitable star orientations, and they apply these results to $(k,r)$-fans to obtain explicit bounds and exact results in several cases, notably for anti-directed $F_{k,r}$. They also explore orientations where the bound $D(n,\vec{H})=2^{\mathrm{ex}(n,H)}$ fails, via bowtie and wheel examples, illustrating limitations and guiding future classification. Overall, the paper extends asymptotic and exact analyses of $D(n,\vec{H})$ beyond edge-critical graphs to important non-edge-critical configurations and highlights the role of structural decompositions in extremal orientation counts.
Abstract
In 1974, Erdős asked the following question: given a graph $G$ and a directed graph $\vec{H}$, how many ways are there to orient the edges of $G$ such that it does not contain $\vec{H}$ as a subgraph? We denote this value by $D(G, \vec{H})$. Further, we let $D(n, \vec{H})$ denote the maximum of $D(G, \vec{H})$ over all graphs $G$ on $n$ vertices. In 2006, Alon and Yuster gave an exact answer when $\vec{H}$ is a tournament. In 2023, Bucić, Janzer, and Sudakov gave asymptotic answers for all directed graphs $\vec{H}$, and in the same paper, they gave an exact answer when $\vec{H}$ is a directed cycle. In this paper, we give a better bound for some specific non-bipartite directed graphs. Further, we obtain exact values of $D(G, \vec{H})$ for some small non-edge-critical directed graphs $\vec{H}$. Finally, for these graphs, we classify all graphs $G$ that attain the bound $D(G, \vec{H}) = D(n, \vec{H})$.