Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.34407

The Plane-Saturation Ratio of Planar Graphs

Qing Cui·Shengjin Ji
View original source ↗Read source document ↗Export metadata ↓

Abstract

For a planar graph \(G\), a plane subgraph \(H\) is plane-saturated with respect to \(G\) if adding any edge to the fixed drawing of \(H\), possibly together with new vertices, either creates a crossing or results in a graph that is not a subgraph of \(G\). The plane-saturation ratio \(\psr(G)\) is the minimum of \(e(H)/e(G)\) over all plane-saturated subgraphs \(H\) of \(G\). For nonnegative integers \(k_1\) and \(k_2\), let \(\cG_{k_1,k_2}\) denote the class of planar graphs in which, for each \(i\in\{1,2\}\), at most \(k_i\) vertices of degree \(i\) have any prescribed neighborhood, where \(k_i=0\) means that vertices of degree \(i\) are forbidden. We prove that every \(G\in\cG_{k_1,k_2}\) with at least one edge satisfies $\psr(G)>\frac{1}{9+k_1+6k_2}.$ This settles two conjectures of Clifton and Salia.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Qing Cui, Shengjin Ji. 2026-10-25. The Plane-Saturation Ratio of Planar Graphs. https://arxiv.org/abs/2610.34407

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Study On The Graph Formulation Of Union Closed Sets Conjecture

The Union Closed Sets Conjecture is one of the most renowned problems in combinatorics. Its appeal lies in the simplicity of its statement contrasted with the potential complexity of its resolution. The conjecture posits that, in any union closed family of sets, there exists at least one element that appears in at least half of the sets within the family. We establish the graph-theoretic version of certain set-theoretic results by connecting the set-based and graph-based formulations. We then prove a theorem in which we investigate the conjecture for graphs, focusing on their decompositions and the position of certain pendant vertices. As a result, we extend the validity of the conjecture to a broader class of graph structures.

math.CO↗
arXiv2026

Embedding edge-colored graphs in expanders with roll-back

We introduce a method to embed edge-colored graphs into families of expander graphs, which generalizes a framework developed by Draganić, Krivelevich, and Nenadov (2022). As an application, we show that each family of sufficiently pseudo-random graphs on $n$ vertices contains every edge-colored subdivision of $K_Δ$, provided that the distance between branch vertices in the subdivision is large enough, the average degree of each graph in the family is at least $(1+o(1))Δ$, and the number of vertices in the subdivision is at most $(1-o(1))n$. This extends the subdivision-embedding results of Draganić, Krivelevich, and Nenadov to the colorful setting. Even in the monochromatic case, our work extends the earlier result, which only applied to graphs with average degree at most $O(n^{1/5})$. This work is motivated in part by the problem of finding structures in distance graphs defined over finite vector spaces. In particular, we obtain nearly spanning subdivisions of complete graphs with arbitrary prescribed nonzero edge distances in sufficiently large subsets of these spaces.

math.CO↗
arXiv2026

Identically vanishing $k$-generalized Fibonacci polynomials

The recurrence for the $k$-Fibonacci polynomials is usually iterated upwards to positive values of $n$ only. When the recurrence is iterated downwards to $n<0$, there are indices where the polynomials vanish identically. This fact does not seem to have been noted in the literature. We derive the set of such indices. We establish the connection of our results to the solution of the Skolem problem for the $k$-Fibonacci numbers. For $k\ge3$ and $n<0$, we show that the degree of the polynomial does not increase monotonically with $|n|$. The so-called `left-justified $k$-nomial triangle' is extended to treat negative indices. We derive expressions for the individual polynomial coefficients (the elementary symmetric polynomials of the roots). We present results for the properties of the polynomials, for both $n>0$ and $n<0$, including factorization of the polynomials and properties of the roots. Results are also derived for real roots. We present new, tighter, bounds on the amplitudes of the nonzero roots. We derive new combinatorial sums for the polynomial coefficients, which are more concise and computationally efficient than previously published expressions.

math.CO↗