Inducing regulation of any digraphs

Abstract

a b s t r a c tFor a given structure D (digraph, multidigraph, or pseudodigraph) and an integer r large enough, a smallest inducing r-regularization of D is constructed. This regularization is an r-regular superstructure of the smallest possible order with bounded arc multiplicity, and containing D as an induced substructure. The sharp upper bound on the number, ρ, of necessary new vertices among such superstructures for n-vertex general digraphs D is determined, ρ being called the inducing regulation number of D. For∆(D) being the maximum among semi-degrees in D, simple n-vertex digraphs D with largest possible ρ are characterized if either r ≥∆(D) or r =∆(D) (where the case r =∆ is not a trivial subcase of r ≥∆).

A partial refining of the Erdős-Kelly regulation

“…Related results on inducing superstructures are presented in [3,10,12,[15][16][17]20]. A generalization of the above Erdős-Kelly result in case of r-regulation (r ≥ ∆ as in König [21,22]) is presented in the authors papers [15][16][17]. For non-inducing regulations (which are described briefly in [17]), see [1,2,[4][5][6][7][8]19].…”