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The authors asymptotically determine the proper rainbow saturation number of the 4-cycle, and provide improved upper bounds for the proper rainbow saturation numbers of the 5-cycle and 6-cycle.

Abstract

The content discusses the study of proper rainbow saturation numbers, which is a recently introduced concept in extremal graph theory. Given a fixed graph F, the proper rainbow saturation number sat*(n, F) is the minimum number of edges in an n-vertex graph that is rainbow F-saturated, meaning it admits a proper edge-coloring that avoids any rainbow copy of F, but the addition of any edge would create such a rainbow copy.
The key results are:
The authors asymptotically determine sat*(n, C4), showing that it is bounded above by 11/6 n + O(1) and below by (11/6 - ε)n for any ε > 0 and sufficiently large n. This separates sat*(n, C4) from the ordinary saturation number sat(n, C4) by a constant multiplicative factor.
The authors provide improved upper bounds on sat*(n, C5) and sat*(n, C6):
sat*(n, C5) ≤ ⌊5n/2⌋ - 4 for n ≥ 9
sat*(n, C6) ≤ 7/3 n + O(1)
The authors establish structural properties of rainbow C4-saturated graphs, showing that any such graph either has a small dominating set or a dominating set with high average degree. This structural insight is crucial for their lower bound on sat*(n, C4).
The authors also discuss general results on the behavior of proper rainbow saturation numbers, including a theorem that for graphs F containing no induced even cycles, sat*(n, F) is linear in n.

Stats

n ≥ 7
sat*(n, C4) ≤ 11/6 n + O(1)
n ≥ 9
sat*(n, C5) ≤ ⌊5n/2⌋ - 4
sat*(n, C6) ≤ 7/3 n + O(1)

Quotes

"We say that an edge-coloring of a graph G is proper if every pair of incident edges receive distinct colors, and is rainbow if no two edges of G receive the same color."
"Given fixed graphs G, F, and a proper edge-coloring c of G, we say that G is rainbow-F-free under c if G does not contain any copy of F which is rainbow with respect to c."
"We call sat*(n, F) the proper rainbow saturation number of F, since all edge-colorings in this setting are proper."

Deeper Inquiries

The proper rainbow saturation numbers of cycles differ from the ordinary saturation numbers of cycles in that they consider edge-colorings that are proper and rainbow. This means that in a rainbow-saturated graph, no two edges share the same color, while in a proper rainbow-saturated graph, every pair of incident edges must have distinct colors. This additional constraint of proper edge-colorings in rainbow saturation numbers can lead to different optimal configurations compared to ordinary saturation numbers.
For other graph families, the behavior of proper rainbow saturation numbers may exhibit significant differences depending on the structural properties of the graphs. Graph families with unique connectivity patterns, specific edge-coloring constraints, or distinct cycle configurations may result in varying proper rainbow saturation numbers compared to their ordinary saturation numbers. Understanding these structural properties and their impact on proper rainbow saturation numbers can provide insights into the behavior of this concept across different graph families.

Beyond the structural properties of rainbow-saturated graphs discussed in the context provided, additional properties that could be leveraged to obtain tighter bounds on proper rainbow saturation numbers for other graph families include:
Symmetry and Regularity: Exploiting symmetrical or regular patterns in the graph structure to optimize proper rainbow saturation numbers.
Coloring Strategies: Developing efficient coloring strategies that minimize the occurrence of rainbow subgraphs in proper rainbow-saturated graphs.
Graph Decomposition: Utilizing graph decomposition techniques to analyze the connectivity and coloring requirements of different components in rainbow-saturated graphs.
Edge Addition Effects: Studying the impact of adding specific edges to rainbow-saturated graphs and how it influences the existence of rainbow subgraphs.
By exploring these and other structural properties of rainbow-saturated graphs, researchers can potentially derive more precise and comprehensive bounds on proper rainbow saturation numbers for a wide range of graph families.

Proper rainbow saturation numbers have applications beyond the initial motivation from additive number theory. Some potential applications and interpretations of proper rainbow saturation numbers include:
Network Security: Proper rainbow saturation numbers can be used to model secure network communication, where each edge represents a secure connection and the absence of rainbow subgraphs indicates secure data transmission.
Resource Allocation: In resource allocation problems, proper rainbow saturation numbers can represent optimal distribution strategies to ensure equitable and efficient resource utilization.
Social Network Analysis: Proper rainbow saturation numbers can be applied to analyze social networks, where edges represent relationships and proper rainbow saturation reflects diverse and non-repetitive interactions among individuals.
Algorithm Design: Proper rainbow saturation numbers can inspire the development of algorithms for graph coloring and optimization, leading to improved solutions in various computational problems.
Exploring these applications and interpretations can provide new insights into the significance and relevance of proper rainbow saturation numbers in diverse fields and problem-solving contexts.

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