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If D = (V, A) is a digraph, its competition graph (with loops) CGl(D) has the vertex set V and {u, v} ⊆ V is an edge of CGl(D) if and only if there is a vertex w ∈ V such that (u, w), (v, w) ∈ A. In CGl(D), loops {v} are allowed only if v is the only predecessor of a certain vertex w ∈ V. For several products D1 ⚬ D2 of digraphs D1 and D2, we investigate the relations between the competition graphs of the factors D1, D2 and the competition graph of their product D1 ⚬ D2.

Properties of digraphs connected with some congruence relations

J. Skowronek-Kaziów (2009)

Czechoslovak Mathematical Journal

The paper extends the results given by M. Křížek and L. Somer, On a connection of number theory with graph theory, Czech. Math. J. 54 (129) (2004), 465–485 (see [5]). For each positive integer $n$ define a digraph $Γ (n)$ whose set of vertices is the set $H = {0, 1, \dots, n - 1}$ and for which there is a directed edge from $a \in H$ to $b \in H$ if $a^{3} \equiv b (mod n) .$ The properties of such digraphs are considered. The necessary and the sufficient condition for the symmetry of a digraph $Γ (n)$ is proved. The formula for the number of fixed points of $Γ (n)$ is established....

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