Bibliographic Information: Buchanan, C., & Rombach, P. (2024). A lower bound on the saturation number and a strengthening for triangle-free graphs. arXiv preprint arXiv:2402.11387v4.
Research Objective: This paper aims to improve the lower bounds on the saturation number of a graph H, denoted sat(n, H), which represents the minimum edge count in an n-vertex graph that doesn't contain H as a subgraph but forms H upon adding any edge. The authors focus on refining these bounds for triangle-free graphs H.
Methodology: The authors introduce two weight functions, wt0 and wt1, based on vertex degrees and neighborhood structures within H. They leverage these functions to analyze the degree distribution in H-saturated graphs, particularly focusing on the degrees of nonadjacent vertex pairs and their implications for the presence of H.
Key Findings: The paper presents two main theorems. Theorem 1 establishes a general lower bound for sat(n, H) for any graph H with no isolated edges, considering the minimum values of wt0 and wt1. Theorem 2 provides a stronger lower bound specifically for triangle-free graphs H, utilizing the property (P) that guarantees a high-degree neighbor for specific low-degree vertex pairs.
Main Conclusions: The authors demonstrate the effectiveness of their bounds by determining the saturation numbers of unbalanced double stars (Ss,t with s < t) up to an additive constant, showing sat(n, Ss,t) = (st + s)n/(2t + 4) + O(1). They also derive new bounds for the saturation numbers of certain diameter-4 caterpillars.
Significance: This research contributes significantly to extremal graph theory by providing improved lower bounds for saturation numbers, particularly for triangle-free graphs. The application of these bounds to determine saturation numbers for specific tree classes highlights their utility and potential for further exploration in graph saturation problems.
Limitations and Future Research: The authors acknowledge that while Theorem 2 strengthens existing bounds for triangle-free graphs, Theorem 1 may not always improve upon previous general bounds, particularly for graphs containing triangles. Future research could explore the role of second neighbors' degrees in refining saturation number bounds for graphs with triangles and larger diameters. Additionally, investigating the applicability of these findings to semisaturation numbers, a related concept in graph theory, could be a promising direction.
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by Calum Buchan... at arxiv.org 10-24-2024
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