When is the Direct Product of Generalized Mycielskians a Cover Graph?

TL;DR

This paper characterizes when the direct product of generalized Mycielskians of graphs results in a cover graph, establishing a condition based on bipartiteness of the original graphs.

Contribution

It provides a necessary and sufficient condition for the direct product of generalized Mycielskians to be a cover graph, linking it to bipartiteness of the original graphs.

Findings

The direct product of generalized Mycielskians is a cover graph iff one of the original graphs is bipartite.

The paper establishes a clear criterion connecting graph properties to cover graph status.

It advances understanding of graph products and cover graph characterization.

Abstract

A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram of a finite partially ordered set. The direct product G X H of graphs G and H is the graph having vertex set V(G) X V(H) and edge set E(G X H) = {(g_i,h_s)(g_j,h_t): g_ig_j belongs to E(G) and h_sh_t belongs to E(H)}. We prove that the direct product M_m(G) X M_n(H) of the generalized Mycielskians of G and H is a cover graph if and only if G or H is bipartite.