Irregular subgraphs

We suggest two related conjectures dealing with the existence of spanning irregular subgraphs of graphs. The first asserts that any d-regular graph on n vertices contains a spanning subgraph in which the number of vertices of each degree between 0 and d deviates from n d+1 by at most 2. The second is that every graph on n vertices with minimum degree δ contains a spanning subgraph in which the number of vertices of each degree does not exceed n δ+1 + 2. Both conjectures remain open, but we prove several asymptotic relaxations for graphs with a large number of vertices n. In particular we show that if d logn ≤ o(n) then every d-regular graph with n vertices contains a spanning subgraph in which the number of vertices of each degree between 0 and d is (1 + o(1)) n d+1 . We also prove that any graph with n vertices and minimum degree δ contains a spanning subgraph in which no degree is repeated more than (1 + o(1)) n δ+1 + 2 times. AMS Subject classification: 05C35, 05C07.

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