Affiliation:
1. Department of Mathematics University of Haifa‐Oranim Haifa Israel
2. Alfréd Rényi Institute of Mathematics Budapest Hungary
3. University of Pannonia Veszprém Hungary
Abstract
AbstractMotivated by the theorem of Győri and Lovász, we consider the following problem. For a connected graph on vertices and edges determine the number of unordered solutions of positive integers such that every is realized by a connected subgraph of with edges. We also consider the vertex‐partition analogue. We prove various lower bounds on as a function of the number of vertices in , as a function of the average degree of , and also as the size of ‐partite connected maximum cuts of . Those three lower bounds are tight up to a multiplicative constant. We also prove that the number of unordered ‐tuples with , that are realizable by vertex partitions into connected parts, is at least .
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