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- W4297009550 abstract "Abstract 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 $frac{n}{d+1}$ by at most $2$ . The second is that every graph on $n$ vertices with minimum degree $delta$ contains a spanning subgraph in which the number of vertices of each degree does not exceed $frac{n}{delta +1}+2$ . 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 $d^3 log n leq o(n)$ 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 $(1+o(1))frac{n}{d+1}$ . We also prove that any graph with $n$ vertices and minimum degree $delta$ contains a spanning subgraph in which no degree is repeated more than $(1+o(1))frac{n}{delta +1}+2$ times." @default.
- W4297009550 created "2022-09-25" @default.
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- W4297009550 date "2022-09-23" @default.
- W4297009550 modified "2023-09-27" @default.
- W4297009550 title "Irregular subgraphs" @default.
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- W4297009550 doi "https://doi.org/10.1017/s0963548322000220" @default.
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