Let q be a prime power, and let m_1,n_1,m_2,n_2 be integers from 1,2,…,q-1. For a monomial digraph D(q;m,n), write D_1=D(q;m_1,n_1) and D_2=D(q;m_2,n_2). The monomial-digraph isomorphism conjecture. The digraphs are isomorphic if and only if there exists an integer k coprime with q-1 such that m_2 equiv k m_1 pmodq-1qquadandqquad n_2 equiv k n_1 pmodq-1. This conjecture gives necessary and sufficient conditions for isomorphism of monomial digraphs and is presented as a related conjecture; the source does not provide a resolution, even for prime q.

The isomorphism criterion for monomial digraphs

The conjecture asserts that two monomial digraphs are isomorphic exactly when their exponent pairs differ by multiplication by a unit modulo $q-1$ . The 2018 paper by Alex Kodess and Felix Lazebnik left the necessity direction open.

Known results

Kodess and Lazebnik (2018) proved the sufficient direction via $(x,y)\mapsto(x^k,y)$ .
They computationally verified necessity for all prime powers $2\le q\le97$ .
They verified the case $m_1=m_2=1$ for odd prime powers $3\le q\le509$ .
They derived independent necessary conditions, but showed these do not imply the conjectured criterion.

October 2026 prime-field result

A preprint by Alexander M. Kodess, Felix Lazebnik, and Mikhail Muzychuk claims to prove the criterion over prime fields and gives counterexamples over extension fields. Thus it would settle the prime-field subproblem while disproving the all-prime-powers formulation, but the preprint is unrefereed and the claims remain unverified.

Current status (as of October 2026): The criterion is claimed proved for prime fields and claimed false over extension fields, but these new results are unverified.

The isomorphism criterion for monomial digraphs

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