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Equitable induced decompositions of twin graphs
Journal article, 2020

problems, we are interested in decomposing graphs with few twin classes into k$induced subgraphs with nearly equal edge numbers, where every edge belongs to exactly one subgraph. Technically we consider the fractional version of the problem, where the vertices of a weighted twin graph can be split into arbitrary fractions, and the k induced subgraphs must receive exactly the same total edge weights. The results then apply to usual graphs, subject to a small discretization error. We show that such equitable induced decompositions are indeed possible for various twin graphs, including all bipartite graphs, cycles Cn and Cn-colorable graphs, and (C3,C5)-free graphs. We also pay attention to the necessary number of vertices (after the splittings) in the induced decompositions. Usually this number is bounded by k+O(1), but for complete bipartite graphs, i.e., when the twin graph is a single edge, roughly 2k^(1/2) vertices suffice, and their exact minimum number is easy to compute for many k.

induced subgraph decomposition

fractional graph theory

equitable partitioning

complex group testing

twins

## Author

### Peter Damaschke

University of Gothenburg

Chalmers, Computer Science and Engineering (Chalmers), Data Science

#### Australasian Journal of Combinatorics

1034-4942 (ISSN) 22023518 (eISSN)

Vol. 76 1 24-40### Roots

Basic sciences

### Subject Categories

Discrete Mathematics