On maximal graphical partitions that are the nearest to a given graphical partition
Vitaly A. Baransky, Tatiana A. Senchonok · Sibirskie Elektronnye Matematicheskie Izvestiya · 2020
A graphical partition is called maximal if it is maximal under domination among graphical partitions of a given weight.Let λ and µ be partitions such that µ ≤ λ.The height of λ over µ is the number of transformations in some shortest sequence of elementary transformations which transforms λ to µ, denoted by height(λ, µ).For a given graphical partition µ, a maximal graphical partition λ such that µ ≤ λ and sum(µ) = sum(λ) is called the h-nearest to µ if it has the minimal height over µ among all maximal graphical partitions λ such that µ ≤ λ and sum(µ) = sum(λ ).The aim is to prove the following result:Let µ be a graphical partition and λ be an h-nearest maximal graphical partition to µ.Then