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Mathematics
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Advances in Graph Labelings and Ramsey Theory in Discrete Structures
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Advances in Graph Labelings and Ramsey Theory in Discrete Structures

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E: Applied Mathematics".

Deadline for manuscript submissions: 22 December 2026 | Viewed by 1136

Editor

Dr. Reza Saei

E-Mail Website
Guest Editor
Faculty of Education and Arts, Nord University, Postbox 1490, 8049 Bodø, Norway
Interests: graph theory; combinatorics and algorithms

Special Issue Information

Dear Colleagues,

Graph labelling and Ramsey theory are well‑established and active areas of discrete mathematics, each with substantial theoretical depth and broad application potential. Since its introduction in the mid‑1960s, graph labelling has developed into a mature field, with more than 3600 publications and over 350 recognized labelling techniques. By assigning labels to vertices, edges, or both under specific combinatorial constraints, graph labelling provides a versatile framework for encoding structural information and revealing underlying patterns in graphs. Its adaptability has led to applications in diverse areas such as coding theory, cryptography, communication networks, circuit and layout design, data management, image processing, and other security systems.

Ramsey theory, originating from Frank P. Ramsey’s seminal work in 1930, studies how large a system must be to guarantee the inevitable emergence of a particular structure. The field has attracted extensive attention, with thousands of contributions examining Ramsey numbers, their combinatorial bounds, and computational complexity. Graph-based Ramsey theory, which naturally intersects with graph labelling and extremal graph theory, remains a central pillar among the many contexts in which Ramsey-type phenomena appear. Research in the field has expanded beyond classical two-colour Ramsey numbers to consider Ramsey numbers defined for other subgraphs, multiple colours, hypergraphs, and ordered structures.

In this Special Issue, we aim to showcase cutting-edge research in graph labelling and Ramsey theory within discrete structures. Contributions spanning theoretical, computational, and applied perspectives are encouraged, particularly those that introduce new methods, solve long-standing problems, or demonstrate applications of discrete structures in related scientific and technological domains.

Potential topics include but are not limited to the following:

  • New graph labelling techniques and theoretical frameworks;
  • Computational and algorithmic approaches to graph labelling problems;
  • Applications of graph labelling;
  • Determining or bounding classical, hypergraph, multicolor, and ordered Ramsey numbers;
  • Ramsey numbers of structured graphs;
  • Anti-Ramsey numbers;
  • Computational and algorithmic approaches to Ramsey theory;
  • Probabilistic, algebraic, topological, and geometric methods in graph labelling and Ramsey problems.

Dr. Reza Saei
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-anonymized peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • graph labelling
  • graph colouring
  • vertex colouring
  • edge colouring
  • ramsey theory
  • ramsey number
  • classical ramsey number
  • discrete structure
  • extremal graph
  • graph class

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Published Papers (2 papers)

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Research

9 pages, 272 KB  
Article
Computational Lower Bounds for Wheel Ramsey Numbers
by Sergey Bereg
Mathematics 2026, 14(13), 2310; https://doi.org/10.3390/math14132310 - 30 Jun 2026
Viewed by 294
Abstract
We study small off-diagonal Ramsey numbers involving wheel graphs. Following Radziszowski’s convention, we write Wn=K1+Cn−1 for the wheel on n vertices. A lower bound for R(Wm,Wn) is [...] Read more.
We study small off-diagonal Ramsey numbers involving wheel graphs. Following Radziszowski’s convention, we write Wn=K1+Cn−1 for the wheel on n vertices. A lower bound for R(Wm,Wn) is certified by a graph G such that G is Wm-free and G¯ is Wn-free. We conduct computational experiments using several complementary search techniques, including partial Ramsey colorings and local search procedures. The computations exploit the structure of wheels: G contains Wn if and only if some vertex neighborhood contains a cycle Cn−1. We report new lower-bound constructions for three off-diagonal wheel Ramsey numbers and provide the corresponding Ramsey graphs as explicit certificates. Full article
(This article belongs to the Special Issue Advances in Graph Labelings and Ramsey Theory in Discrete Structures)
29 pages, 2318 KB  
Article
From Cell-Specific Heuristics to Transferable Structural Search for Ramsey Graph Construction
by Sorin Liviu Jurj
Mathematics 2026, 14(8), 1367; https://doi.org/10.3390/math14081367 - 19 Apr 2026
Viewed by 427
Abstract
Recent automated search methods have improved lower bounds for several Ramsey numbers, but the strongest gains often depend on structured seeding and cell-specific heuristic discovery. This leaves open a more fundamental question: Can a useful search structure be transferred across related Ramsey cells [...] Read more.
Recent automated search methods have improved lower bounds for several Ramsey numbers, but the strongest gains often depend on structured seeding and cell-specific heuristic discovery. This leaves open a more fundamental question: Can a useful search structure be transferred across related Ramsey cells rather than rediscovered independently for each target instance? This work proposes a teacher–student framework for transferable structural search in Ramsey graph construction, inspired by the structure-distillation logic of Physics Structure-Informed Neural Networks (Ψ-NNs). The framework builds compressed structural representations from teacher witnesses and search traces, extracts reusable motifs and relations, and reconstructs transfer candidates. These are refined by balanced search and, for weak R(3, s) cells, by exact small-cell supervision. The framework is evaluated as a proof of concept across five Ramsey cells under transfer, matched-compute, search, ablation, and interpretability settings, including a proportional shift-scaling baseline and a greedy triangle-closing baseline that probe the structure-validity frontier from complementary directions. Supplementary experiments cover seed robustness, budget sensitivity, transfer-neighborhood variation, structural-resolution changes, stronger exact supervision, cross-r teacher pooling, single-teacher configurations, and scaling behavior across graph sizes. The results show that the portfolio version of the framework is the strongest balanced transfer method in the current study, while a structure-dominant oracle achieves stronger witness-shape agreement but worse Ramsey-valid construction. These findings reveal a clear structure-validity frontier and suggest that transferable Ramsey search should be evaluated by how well structural priors survive the validity constraints of new cells. Full article
(This article belongs to the Special Issue Advances in Graph Labelings and Ramsey Theory in Discrete Structures)
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