On p-competition graphs of loopless Hamiltonian digraphs without symmetric arcs and graph operations

Abstract

For a digraph $D$, the $p$-competition graph $C_{p}(D)$ of $D$ is the graph satisfying the following: $V(C_{p}(D))=V(D)$, for $x,y \in V(C_{p}(D))$, $xy \in E(C_{p}(D))$ if and only if there exist distinct $p$ vertices $v_{1},$ $v_{2},$ $...,$ $v_{p}$ $\in$ $V(D)$ such that $x \rightarrow v_{i}$, $y \rightarrow v_{i}$ $\in$ $A(D)$ for each $i=1,2,$ $...,$ $p$. We show the $H_1 \cup H_2$ is a $p$-competition graph of a loopless digraph without symmetric arcs for $p \geq 2$, where $H_1$ and $H_2$ are $p$-competition graphs of loopless digraphs without symmetric arcs and $V(H_1) \cap V(H_2)$ $=$ $\{ \alpha \}$. For $p$-competition graphs of loopless Hamiltonian digraphs without symmetric arcs, we obtain similar results. And we show that a star $K_{1,n}$ is a $p$-competition graph of a loopless Hamiltonian digraph without symmetric arcs if $n \geq 2p+3$ and $p \geq 3$. Based on these results, we obtain conditions such that spiders, caterpillars and cacti are $p$-competition graphs of loopless digraphs without symmetric arcs. We also obtain conditions such that these graphs are $p$-competition graphs of loopless Hamiltonian digraphs without symmetric arcs.