Degree-restricted semi-saturation numbers of cliques and its applications

Abstract: A graph $G$ is said to be $F$-semi-saturated if the addition of any nonedge $e \not \in E(G)$ would create a new copy of $F$ in $G+e$. The semi-saturation number $ssat(n,F)$ is the minimum number of edges in an $F$-semi-saturated graph of order $n$. In this paper we investigate the semi-saturation number of $K_r$ on $n$ vertices with maximal degree at most $Δ$, denoted by $ssat<sup>Δ(n,K_r)$. This investigation was suggested by Erd\H os, Rényi and Sós, who in 1966 considered the graph of diameter 2 with degree restrictions, equivalently $ssat<sup>Δ(n,K_3)$. The following are some of our results. For arbitrary $r \geq 4$, we show that the limit $$\lim_{n \rightarrow \infty} ssat<sup>{cn}(n,K_r)/n$$ exists for all $0 &lt; c \leq 1$, except for some sparse values of $c$ contained in a countable and rational sequence $c_i \rightarrow 0$. Moreover, we establish the asymptotic behaviour of this limit for $\frac{r}{r+2} &lt; c &lt;1$ and determine the exact value of $ssat<sup>Δ(n,K_r)$ for some specific $Δ$. As an application, we determine the relation between the saturation number of the join graph $K_r \vee F$ and that of $F$ for a large class of pairs $(r,F)$.

• The paper establishes the asymptotic behavior of degree-restricted semi-saturation numbers for cliques using explicit piecewise linear functions.
• It reformulates the problem as a linear program over k-intersecting hypergraphs, revealing sharp thresholds and rational discontinuities.
• The work applies these results to derive exact saturation numbers for graph joins, advancing our understanding of extremal configurations.

Degree-Restricted Semi-Saturation Numbers for Cliques and Applications

Introduction and Motivation

This paper addresses the classical problem of graph semi-saturation with additional constraints on maximum vertex degree, focusing on the semi-saturation number for cliques, denoted $ssat^{\Delta}(n,K_r)$. For a graph $G$ on $n$ vertices and a fixed graph $F$ (here, a clique $K_r$), $G$ is $F$-semi-saturated if the addition of any nonedge creates a new copy of $F$. The semi-saturation number is the minimal number of edges in such a graph, and with maximum degree at most $\Delta$, the degree-restricted number becomes $ssat^{\Delta}(n,K_r)$. This problem generalizes classic results on the minimum size of saturated graphs, incorporating additional structural restrictions relevant for extremal and probabilistic combinatorics.

Erdős, Rényi, and Sós (1966) originally considered the $G$0 case (triangle) with degree constraints, but the asymptotic and explicit behavior for larger cliques ($G$1) remained unresolved. This paper addresses these open questions, characterizing the limiting and exact behavior of $G$2 for various regimes of the degree proportion $G$3 and linking these results to saturation numbers of certain graph joins.

Main Results

Asymptotic Regimes and Explicit Limits

The authors show that for all $G$4 and almost all $G$5, the limit

$G$6

exists and can be identified as a piecewise linear function $G$7, except at points $G$8 belonging to a specific rational sequence tending to 0. The function $G$9 is explicitly determined for all $n$0 in the range $n$1. The discontinuities in $n$2 occur at a countable, sparse set of rational values.

Critically, they prove:

• Existence and piecewise linearity: $n$3 is monotone nonincreasing, piecewise linear, and right-continuous with all breakpoints rational and determined by properties of $n$4-intersecting hypergraphs (where $n$5).
• Interval-wise explicit formulas: On each maximal continuity interval, $n$6 is given by a linear expression whose coefficients depend on $n$7 and $n$8. Specifically,

$n$9

• Attainability: For each interval, constructions are given that match the lower bound, demonstrating sharpness.

Linear Programming and Hypergraph Connection

A significant conceptual advance is the formulation of the extremal function $F$0 as an explicit linear program over $F$1-intersecting hypergraphs, linked to the notion of fractional matchings. This transforms the semi-saturation problem into a fractional, combinatorial optimization problem:

• LP formulation: $F$2 is the infimum of a weighted edge-size sum subject to fractional coverage and packing constraints, over all $F$3-intersecting hypergraphs with prescribed fractional matching properties.
• Limiting structure: The convexity, continuity, and finiteness of breakpoints in $F$4 are established via properties of these LPs.

Exact Results for Special Maximum Degree

The authors also obtain exact formulas for $F$5 for maximum degree $F$6 in certain discrete ranges. For example, for $F$7 (excluding $F$8), they prove: $F$9 and identify the extremal constructions.

Applications to Saturation Numbers of Graph Joins

A key application is to the $K_r$0-join problem: determining $K_r$1 in terms of $K_r$2 for a large class of pairs $K_r$3. They settle many previously open cases, proving that if $K_r$4 is a graph (without isolated vertices, with every edge in a $K_r$5) and $K_r$6 is sufficiently small, then

$K_r$7

for all large $K_r$8. The method shows that every extremal graph in this case has $K_r$9 conical vertices, and the result generalizes known cases for paths, cycles, and linear forests. The reduction is enabled by the semi-saturation results proved in this work.

Numerical and Structural Highlights

• Sharp limiting constants for the degree-restricted semi-saturation number for all $G$0, with explicit constructions.
• Rigorous identification of breakpoints, all rational, with verified sharpness via matching lower and upper bounds.
• Inheritance of extremality: results for join graphs imply existence of conical vertices in all extremal configurations under natural graph-theoretic assumptions.

Implications and Future Directions

These results advance the fine-grained understanding of saturation-type extremal functions under degree constraints, resolving long-standing combinatorial questions. The identification of $G$1 via fractional LPs over hypergraphs creates a direct bridge between extremal graph theory and fractional combinatorial optimization. Further, the explicit interval structure and construction techniques suggest routes for investigating related constrained extremal functions, particularly in the context of random or sparse graph regimes.

Some potential directions include:

• Extension to other monotone properties (beyond cliques), particularly forbidden substructures in sparse regimes.
• Algorithmic aspects: optimization of explicit extremal constructions for use in probabilistic or network design.
• Broader LP approaches: leveraging fractional hypergraph dualities in other areas of extremal combinatorics.

Conclusion

This paper provides a detailed characterization of the degree-restricted semi-saturation number for cliques, both asymptotically and exactly in key ranges. Through linear programming over intersecting hypergraphs and explicit graph-theoretic constructions, the authors close fundamental open problems and extend the reach of saturation theory to new classes of graph joins. The methods and results will serve as a basis for further study of degree-constrained extremal problems and their interconnections with fractional and probabilistic combinatorics.

For further technical depth and the proofs of these results, see "Degree-restricted semi-saturation numbers of cliques and its applications" (2606.28727).