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Irregular subgraphs

Part of: Graph theory

Published online by Cambridge University Press:  23 September 2022

Noga Alon  and
Fan Wei
Noga Alon
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA Schools of Mathematics and Computer Science, Tel Aviv University, Tel Aviv 69978, Israel
Fan Wei*
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA
*
*Corresponding author. Email: fanw@princeton.edu
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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.

Keywords

irregular subgraphrepeated degrees

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05C07: Vertex degrees
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 32 , Issue 2 , March 2023 , pp. 269 - 283
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

†

Research supported in part by NSF grant DMS-2154082 and BSF grant 2018267.

‡

Research supported by NSF Award DMS-1953958.

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