Moon-type theorems on circuits in strongly connected tournaments of order $N$ and diameter $D$

Abstract

Let $T$ be a strongly connected tournament of order $n\ge 4$ whose diameter does not exceed $d\ge 3.$ Denote by $c_{\ell}(T)$ the number of circuits of length $\ell$ in $T.$ In our recent paper, we construct a strongly connected tournament $T_{d,n}$ of order $n$ with diameter $d$ and conjecture that $c_{\ell}(T)\ge c_{\ell}(T_{d,n})$ for any $\ell=3,...,n.$ In particular, for $d=n-1,$ this inequality is true and yields the known Moon (lower) bound $c_{\ell}(T)\ge n-\ell+1.$ Moreover, we suggest that if $n+3\le 2d,$ then for any given $\ell$ taken in the range $n-d+3,...,d,$ the equality $c_{\ell}(T)=c_{\ell}(T_{d,n})$ implies that $T$ is isomorphic to $T_{d,n}$ or its converse $T_{d,n}^{-}.$ For $d=n-1,$ the corresponding particular statement is nothing else than Las Vergnas' theorem. Recently, we have confirmed the posed conjecture for the case $d=n-2.$ In the present paper, we show that it is also true for $d=n-3.$