Additive combinatorial designs

Abstract A $$2-(v, k, \lambda )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>-</mml:mo> <mml:mo>(</mml:mo> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> design is additive if, up to isomorphism, the point set is a subset of an abelian group G and every block is zero-sum. This definition was introduced in Caggegi et al. (J Algebr Comb 45:271-294, 2017) and was the starting point of an interesting new theory. Although many additive designs have been constructed and known designs have been shown to be additive, these structures seem quite hard to construct in general, particularly when we look for additive Steiner 2-designs. One might generalize additive Steiner 2-designs in a natural way to graph decompositions as follows: given a simple graph $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> , an additive $$(K_v,\Gamma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>v</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>Γ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -design is a decomposition of the graph $$K_v$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>v</mml:mi> </mml:msub> </mml:math> into subgraphs ( blocks ) $$B_1,\dots ,B_t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> </mml:math> all isomorphic to $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> , such that the vertex set $$V(K_v)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>v</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is a subset of an abelian group G , and the sets $$V(B_1), \dots , V(B_t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:mi>V</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> are zero-sum in G . In this work we begin the study of additive $$(K_v,\Gamma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>v</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>Γ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -designs: we develop different tools instrumental in constructing these structures, and apply them to obtain some infinite classes of designs and many sporadic examples. We will consider decompositions into various graphs $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> , for instance cycles, paths, and k -matchings. Similar ideas will also allow us to present here a sporadic additive 2-(124, 4, 1) design.