The minimum rank problem over the finite field of order 2: minimum rank 3

Details The minimum rank problem over the finite field of order 2: minimum rank 3 The minimum rank problem over the finite field of order 2: minimum rank 3

2006-12-12

arxiv.org/abs/math/0612331v3

Mathematics

Combinatorics

40 pages, added Sage program and improvements from referee process

Scientific paper

Our main result is a sharp bound for the number of vertices in a minimal forbidden subgraph for the graphs having minimum rank at most 3 over the finite field of order 2. We also list all 62 such minimal forbidden subgraphs. We conclude by exploring how some of these results over the finite field of order 2 extend to arbitrary fields and demonstrate that at least one third of the 62 are minimal forbidden subgraphs over an arbitrary field for the class of graphs having minimum rank at most 3 in that field.

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