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New bounds on maximal linkless graphs

Ramin Naimi, Andrei Pavelescu and Elena Pavelescu

Algebraic & Geometric Topology 23 (2023) 2545–2559

We construct a family of maximal linklessly embeddable graphs on n vertices and 3n 5 edges for all n 10, and another family on n vertices and m < 25 12n 1 4 edges for all n 13. The latter significantly improves the lowest edge-to-vertex ratio for any previously known infinite family. We construct a family of graphs showing that the class of maximal linklessly embeddable graphs differs from the class of graphs that are maximal without a K6 minor studied by L Jørgensen. We give necessary and sufficient conditions for when the clique sum of two maximal linklessly embeddable graphs over K2, K3 or K4 is a maximal linklessly embeddable graph, and use these results to prove our constructions yield maximal linklessly embeddable graphs.

maximal linkless graphs, clique sums
Mathematical Subject Classification
Primary: 57M15
Secondary: 05C10
Received: 19 September 2020
Revised: 28 December 2021
Accepted: 18 January 2022
Published: 7 September 2023
Ramin Naimi
Department of Mathematics
Occidental College
Los Angeles, CA
United States
Andrei Pavelescu
Mathematics and Statistics Department
University of South Alabama
Mobile, AL
United States
Elena Pavelescu
Mathematics and Statistics Department
University of South Alabama
Mobile, AL
United States

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