Odd Covers of Complete Graphs and Hypergraphs
Imre Leader, Ta Sheng Tan

TL;DR
This paper proves a conjecture about the odd cover number of complete graphs, extends results to hypergraphs, and provides new constructions and bounds for various cases, advancing understanding of odd covers in graph theory.
Contribution
It confirms the conjecture that the odd cover number of complete graphs is always (n+1)/2 for odd n and provides exact values and bounds for hypergraphs and even n cases.
Findings
Odd cover number of K_n is (n+1)/2 for odd n.
Odd cover number of K_n^{(3)} is n/2 for even n.
For r=3,4, the odd cover number is less than the partition number.
Abstract
The `odd cover number' of a complete graph is the smallest size of a family of complete bipartite graphs that covers each edge an odd number of times. For odd, Buchanan, Clifton, Culver, Nie, O'Neill, Rombach and Yin showed that the odd cover number of is equal to or , and they conjectured that it is always . We prove this conjecture. For even, Babai and Frankl showed that the odd cover number of is always at least , and the above authors and Radhakrishnan, Sen and Vishwanathan gave some values of for which equality holds. We give some new examples. Our constructions arise from some very symmetric constructions for the corresponding problem for complete hypergraphs. Thus the odd cover number of the complete 3-graph is the smallest number of complete 3-partite 3-graphs such that each 3-set is in an odd number of…
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Graph theory and applications
