Academic Journal
On (r,c)-constant, planar and circulant graphs
| Title: | On (r,c)-constant, planar and circulant graphs |
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| Authors: | Caro, Yair, Mifsud, Xandru |
| Publisher Information: | Uniwersytet Zielonogorski * Wydzial Matematyki, Informatyli i Ekonometrii,Technical University Zielona Gora, Institute of Mathematics |
| Publication Year: | 2025 |
| Collection: | University of Malta: OAR@UM / L-Università ta' Malta |
| Subject Terms: | Graph theory, Mathematics -- Charts, diagrams, etc, Graph algorithms, Combinatorial analysis, Mathematical statistics -- Data processing |
| Description: | This paper concerns (r, c)-constant graphs, which are r-regular graphs in which the subgraph induced by the open neighbourhood of every vertex has precisely c edges. The family of (r, c)-graphs contains vertex-transitive graphs (and in particular Cayley graphs), graphs with constant link (sometimes called locally isomorphic graphs), (r, b)-regular graphs, strongly regular graphs, and much more. This family was recently introduced in [6], serving as an important tool in constructing flip graphs [6, 14]. In this paper we shall mainly deals with the following. (i) Existence and non-existence of (r, c)-planar graphs. We completely determine the cases of existence and non-existence of such graphs and supply the smallest order in the case when they exist. (ii) We consider the existence of (r, c)-circulant graphs. We prove that for c ≡ 2 (mod 3) no (r, c)-circulant graph exists and that for c ≡ 0, 1 (mod 3), c > 0 and r ≥ 6 + √8c−5/3 there exist (r, c)-circulant graphs. Moreover for c = 0 and r ≥ 1, (r, 0)-circulants exist. (iii) We consider the existence and non-existence of small (r, c)-constant graphs, supplying a complete table of the smallest order of graphs we found for 0 ≤ c ≤ C(r, 2) and r ≤ 6. We shall also determine all the cases in this range for which (r, c)-constant graphs do not exist. We establish a public database of (r, c)-constant graphs for varying r, c and order. ; peer-reviewed |
| Document Type: | article in journal/newspaper |
| Language: | English |
| Relation: | https://www.um.edu.mt/library/oar/handle/123456789/140903 |
| DOI: | 10.7151/dmgt.2551 |
| Availability: | https://www.um.edu.mt/library/oar/handle/123456789/140903 https://doi.org/10.7151/dmgt.2551 |
| Rights: | info:eu-repo/semantics/openAccess ; The copyright of this work belongs to the author(s)/publisher. The rights of this work are as defined by the appropriate Copyright Legislation or as modified by any successive legislation. Users may access this work and can make use of the information contained in accordance with the Copyright Legislation provided that the author must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the prior permission of the copyright holder. |
| Accession Number: | edsbas.5AFB86A5 |
| Database: | BASE |
| DOI: | 10.7151/dmgt.2551 |
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