Degree- and Orbit-Balanced Γ-Designs When Γ Has Five Vertices

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Abstract

A Γ-design of the complete graph math formula is a set math formula of subgraphs isomorphic to Γ (blocks) whose edge-sets partition the edge-set of math formula. math formula is balanced if the number of blocks containing x is the same number of blocks containing y for any two vertices x and y. math formula is orbit-balanced, or strongly balanced, if the number of blocks containing x as a vertex of a vertex-orbit A of Γ is the same number of blocks containing y as a vertex of A, for any two vertices x and y and for every vertex-orbit A of Γ. We say that math formula is degree-balanced if the number of blocks containing x as a vertex of degree d in Γ is the same number of blocks containing y as a vertex of degree d in Γ, for any two vertices x and y and for every degree d in Γ. An orbit-balanced Γ-design is also degree-balanced; a degree-balanced Γ-design is also balanced. The converse is not always true. We study the spectrum for orbit-balanced, degree-balanced, and balanced Γ-designs of math formula when Γ is a graph with five vertices, none of which is isolated. We also study the existence of balanced (respectively, degree-balanced) Γ-designs of math formula which are not degree-balanced (respectively, not orbit-balanced).

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