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 Prikl. Diskr. Mat. Suppl., 2018, Issue 11, Pages 16–20 (Mi pdma380)  Theoretical Foundations of Applied Discrete Mathematics

An improved formula for the universal estimation of digraph exponents

V. M. Fomichevabc

a Financial University under the Government of the Russian Federation, Moscow
b National Engineering Physics Institute "MEPhI", Moscow
c Federal Research Center "Computer Science and Control" of Russian Academy of Sciences, Moscow

Abstract: An early formula by A. L. Dulmage and N. S. Mendelsohn (1964) for the universal estimation of $n$-vertex primitive digraph exponent is based on a system $\hat C=\{C_1,…,C_m\}$ of directed circuits in the graph with lengths $l_1,…,l_m$ respectively such that $\mathrm{gcd}(l_1,…,l_m)=1$. A new formula is based on a similar circuit system $\hat C$ with $\mathrm{gcd}(l_1,…,l_m)=d\geq1$. Also, the new formula uses the values $r_{i,j}^{s/d}(\hat C)$ that are the lengths of the shortest paths from a vertex $i$ to a vertex $j$ going through the circuit system $\hat C$ and having the length comparable to $s$ modulo $d$, $s\in\{0,…,d-1\}$. It's shown, that $\exp\Gamma\leq1+\hat F(L(\hat C))+R(\hat C)$ where $\hat F(L)=d\cdot F(l_1/d,…,l_m/d)$ and $F(a_1,…,a_m)$ is the Frobenius number, $R(\hat C)=\max_{(i,j)}\max_s\{r_{i,j}^{s/d}(\hat C)\}$. For a class of $2k$-vertex primitive digraphs, it is proved that the improved formula gives the value of estimation $2k$, but the early formula gives the value of estimation $3k-2$.

Keywords: Frobenius number, primitive graph, exponent of graph.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00226

DOI: https://doi.org/10.17223/2226308X/11/5  Full text: PDF file (627 kB) References: PDF file   HTML file

UDC: 519.1

Citation: V. M. Fomichev, “An improved formula for the universal estimation of digraph exponents”, Prikl. Diskr. Mat. Suppl., 2018, no. 11, 16–20 Citation in format AMSBIB
\Bibitem{Fom18} \by V.~M.~Fomichev \paper An improved formula for the universal estimation of digraph exponents \jour Prikl. Diskr. Mat. Suppl. \yr 2018 \issue 11 \pages 16--20 \mathnet{http://mi.mathnet.ru/pdma380} \crossref{https://doi.org/10.17223/2226308X/11/5} 

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