--- title: Maximum Uncrossed Subgraph Number url: https://www.emergentmind.com/topics/maximum-uncrossed-subgraph-number type: topic --- # Maximum Uncrossed Subgraph Number The **maximum uncrossed subgraph number** is a graph-drawing parameter that measures, for a single drawing of a graph \(G\) in the plane, how many edges can be made uncrossed simultaneously. In the notation used in recent work, it is denoted \(h(G)\) and defined as the maximum number of edges of \(G\) that are not crossed in a drawing of \(G\) in the plane [2507.20937]. This parameter is complementary to the edge crossing number, which minimizes the number of edges that are crossed at least once, and it is closely linked to the **uncrossed number** \(\mathrm{unc}(G)\), which asks for the minimum number of drawings needed so that every edge of \(G\) is uncrossed in at least one drawing [2407.21206]. A distinct but related meaning also occurs in fixed-order book embeddings, where the objective is the maximum number of edges that can be kept while partitioning them into \(k\) noncrossing pages under a prescribed vertex order [1504.05908]. ## 1. Definitions and parameter landscape An **uncrossed collection** of drawings of a graph \(G=(V,E)\) is a collection \(\mathcal{D}=\{D_1,\dots,D_k\}\) such that for every edge \(e\in E\), there exists some drawing \(D_i\in\mathcal{D}\) in which \(e\) is not crossed [2306.09550]. The **uncrossed number** \(\mathrm{unc}(G)\) is the least cardinality \(k\) of such a collection [2407.21206]. This shifts attention from a single optimal drawing to a multi-view representation in which each edge appears cleanly at least once. Within this framework, the maximum uncrossed subgraph number is the one-drawing parameter \[ h(G):=\max_D \left|\{e\in E(G): e\text{ is uncrossed in }D\}\right|, \] where the maximum ranges over drawings \(D\) of \(G\) in the plane [2407.21206]. If \(m=|E(G)|\), then the complement \(m-h(G)\) is the minimum possible number of crossed edges in a drawing, i.e. the edge crossing number viewpoint [2407.21206]. The two perspectives are therefore equivalent at the level of a single drawing: maximizing uncrossed edges is the same optimization as minimizing crossed edges. The relation between \(h(G)\) and \(\mathrm{unc}(G)\) is immediate. Since one drawing can contribute at most \(h(G)\) uncrossed edges, any uncrossed collection must satisfy \[ \mathrm{unc}(G)\ge \left\lceil \frac{m}{h(G)}\right\rceil \] [2507.20937]. The same literature places \(\mathrm{unc}(G)\) between classical covering parameters: \[ \theta(G)\le \mathrm{unc}(G)\le \theta_o(G)\le 2\theta(G), \] where \(\theta(G)\) is thickness and \(\theta_o(G)\) is outerthickness [2306.09550]. This does not directly determine \(h(G)\), but it situates the parameter inside the broader theory of planar and outerplanar edge decompositions. A useful structural notion is that of an **uncrossed subdrawing**. In a drawing \(D\), the edges that are uncrossed form a planar subdrawing, and the cited structural lemma states that each edge of the original graph has its endpoints on a common face of that subdrawing [2407.21206]. This face-incidence property is stronger than mere planarity and is central to the best known general bounds on \(h(G)\). ## 2. General bounds for \(h(G)\) For a connected graph \(G\) with \(n\ge 3\) vertices and \(m\) edges, the strongest general bound stated directly in terms of the maximum uncrossed subgraph number is \[ h(G)\le 3n-6-\sqrt{2m}+\sqrt{6(n-2)} \] [2507.20937]. This is the main theorem of the 2025 bound paper and is presented there as the central result in the \(h(G)\)-language. Through the inequality \(\mathrm{unc}(G)\ge \lceil m/h(G)\rceil\), it yields the corresponding lower bound \[ \mathrm{unc}(G)\ge \left\lceil \frac{m}{3n-6-\sqrt{2m}+\sqrt{6(n-2)}}\right\rceil \] for connected graphs [2507.20937]. For dense graphs with \[ m=\varepsilon n^2, \] the same theorem specializes to the asymptotic estimate \[ h(G)\le (3-\sqrt{2\varepsilon})n+o(n) \] [2507.20937]. The coefficient \[ c'_\varepsilon=3-\sqrt{2\varepsilon} \] also appears in the dense-graph form of the uncrossed-number bound. As \(\varepsilon\) increases, the upper bound on the number of simultaneously uncrossed edges decreases linearly in \(n\), quantifying the fact that in dense graphs only a vanishing fraction of the total \(m=\Theta(n^2)\) edges can remain uncrossed in any one drawing. These results refine earlier general estimates. A trivial lower bound on \(\mathrm{unc}(G)\) comes from thickness: \[ \left\lceil\frac{|E(G)|}{3|V(G)|-6}\right\rceil\le \mathrm{unc}(G), \] because any planar subgraph on \(n\) vertices has at most \(3n-6\) edges [2507.20937]. An earlier general lower bound for connected graphs was stated in the form \[ \mathrm{unc}(G)\ge \left\lceil \frac{m}{(3n-5+\sqrt{(3n-5)^2-4m})/2}\right\rceil, \] which, for dense graphs, was summarized as \[ \mathrm{unc}(G)\ge \left\lceil \frac{m}{c_\varepsilon n}\right\rceil,\qquad c_\varepsilon\ge 2.82 \] [2507.20937]. By the inequality \(\mathrm{unc}(G)\ge m/h(G)\), this earlier result implicitly gives a bound of the form \(h(G)\le c_\varepsilon n\) up to lower-order terms, whereas the newer bound improves the dense-regime constant to \(3-\sqrt{2\varepsilon}\) [2507.20937]. The same paper also states triangle-free analogues of these estimates, indicating that forbidding triangles sharpens the face-counting argument and improves the constants for special graph classes [2507.20937]. ## 3. Proof ideas and structural method The proof strategy for the bound on \(h(G)\) is based on choosing an uncrossed subdrawing \(D'\) of a drawing \(D\) and analyzing the planar combinatorics of \(D'\) [2507.20937]. Because \(D'\) is planar, its faces control where the remaining crossed edges of \(G\) can be routed: every edge of \(G\) has both endpoints on a common face of \(D'\) [2407.21206]. This converts the problem into a constrained counting question about how many additional edges can be “supported” by the faces of a planar subdrawing. The analysis uses the face set \(\mathcal{F}\), the face lengths \(|F|\), the number \(v(F)\) of vertices on a face, and the numbers \(s_\ell\) of faces of length \(\ell\) [2507.20937]. Standard planar identities enter at this point. In particular, a planar drawing satisfies \[ \sum_{F\in\mathcal{F}} |F| = 2m_i, \] where \(m_i\) is the number of edges in the uncrossed subdrawing, and Euler’s formula gives \[ n-m_i+f_i=2 \] for connected planar subdrawings [2507.20937]. Additional inequalities on weighted sums of face counts then restrict how many crossed edges can be associated with short faces. The paper derives two complementary estimates: a “simple bound,” obtained from Euler’s formula and bounded face lengths, and a “complex bound,” obtained from a more precise accounting of how many extra edges can lie inside each face [2507.20937]. These are combined through a parameter \(k\), then optimized by choosing \(k\) as a function of the graph density. The final step selects \[ \alpha=\sqrt{(3n-6)/m} \] to obtain the closed-form upper bound \[ h(G)\le 3n-6-\sqrt{2m}+\sqrt{6(n-2)} \] [2507.20937]. Conceptually, the argument is a refinement of the naive observation “uncrossed edges form a planar graph”: it uses not only planarity, but also the detailed face structure forced by uncrossedness. ## 4. Tightness, extremal constructions, and dense-regime behavior The bound \[ h(G)\le 3n-6-\sqrt{2m}+\sqrt{6(n-2)} \] is exact for planar triangulations. If \(G\) is a planar triangulated graph with \(n\) vertices and \(m=3n-6\) edges, then \(G\) has a drawing with all edges uncrossed, so \(h(G)=3n-6\), and substituting \(m=3n-6\) into the bound gives equality [2507.20937]. More significantly, the 2025 paper provides an asymptotically tight construction for arbitrary dense regimes below the complete-graph limit. It states that for any density \[ 0<\varepsilon\le \frac12-\frac{1}{2n} \quad\text{and}\quad n\ge \frac{3}{\varepsilon}, \] there exists a connected graph \(G\) on \(n\) vertices and \(m\) edges such that \[ h(G)\ge 3n-3-\sqrt{2m} \] and \[ \varepsilon \le \frac{m}{n^2}\le \varepsilon+\frac1n+\frac{1}{2n^2} \] [2507.20937]. Since the difference between the general upper bound and this lower bound is at most \[ \sqrt{6(n-2)}-3=o(n), \] the theorem shows that the upper bound is tight up to additive lower-order terms on dense graphs for all fixed \(\varepsilon>0\) below \(1/2\) [2507.20937]. The construction begins with a wheel \(W_{x+1}\), drawn planarly, and then adds all possible edges between the outer-cycle vertices to form a clique \(K_x\) inside the outer face, with those added edges crossing each other [2507.20937]. The remaining \(n-x-1\) vertices are inserted one by one into triangular faces of the planar part and connected to the three vertices of the chosen triangle, thereby preserving a triangulated interior. In the notation stated in the construction, \[ m'=3n-3-x \le h(G_{x,n}) \] and \[ m=3n-3+\frac{x(x-5)}{2}. \] Since \(\sqrt{2m}\ge x\), the construction yields \[ h(G_{x,n})\ge 3n-3-\sqrt{2m} \] [2507.20937]. For complete graphs, the dense-regime asymptotics become especially transparent. The paper notes that the general bound is tight up to low-order terms for \(\varepsilon\approx 1/2\), “as warranted by complete graphs” [2507.20937]. This matches the exact value \[ h(K_n)=2n-2, \] which is the complete-graph case of the classical one-drawing maximum [2407.21206]. Since \(K_n\) has density asymptotically \(1/2\), the coefficient \(3-\sqrt{2\varepsilon}\) becomes \(2\), agreeing with the leading term of \(h(K_n)\). ## 5. Exact values for classical graph families The exact value \[ h(K_n)=2n-2 \] for complete graphs plays a central role in the theory [2407.21206]. It supplies both a benchmark for tightness and the lower-bound ingredient used in exact uncrossed-number formulas for complete graphs. The corresponding uncrossed number is \[ \mathrm{unc}(K_n)= \begin{cases} \left\lceil \dfrac{n+1}{4}\right\rceil,& n\notin\{4,7\},\\[4pt] 3,& n=7,\\[4pt] 1,& n=4, \end{cases} \] showing how the one-drawing constraint encoded by \(h(K_n)\) propagates to the multi-drawing covering problem [2407.21206]. For complete bipartite graphs, Mengersen’s exact determination of the one-drawing parameter is \[ h(K_{m,n})= \begin{cases} 2m+n-2,& m=n,\\[4pt] 2m+n-1,& m