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1. Lombardi drawings of knots and links

   https://doi.org/10.1007/978-3-319-73915-1_10  

   Kindermann, P ; Kobourov, S ; Loeffler, M ; Noellenburg, M ; Schulz, A. ; Vogtenhuber, B ( January 2019 , Journal of computational geometry)

   Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into lR2, such that no more than two points project to the same point in lR2. These diagrams are drawings of 4-regular plane multigraphs. Knots are typically smooth curves in lR3, so their projections should be smooth curves in lR2 with good continuity and large crossing angles: exactly the properties of Lombardi graph drawings (defned by circular-arc edges and perfect angular resolution). We show that several knots do not allow crossing-minimal plane Lombardi drawings. On the other hand, we identify a large class of 4-regular plane multigraphs that do have plane Lombardi drawings. We then study two relaxations of Lombardi drawings and show that every knot admits a crossing-minimal plane 2-Lombardi drawing (where edges are composed of two circular arcs). Further, every knot is near-Lombardi, that is, it can be drawn as a plane Lombardi drawing when relaxing the angular resolution requirement by an arbitrary small angular offset ε, while maintaining a 180◦ angle between opposite edges. 

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2. Simple Topological Drawings of k-Planar Graphs.

   https://doi.org/10.1007/978-3-030-68766-3_31  

   Hoffmann, Michael ; Liu, Chih-Hung ; Reddy, Meghana M. ; Toth, Csaba D. ( February 2021 , Graph Drawing and Network Visualization)

   null (Ed.)

   Every finite graph admits a simple (topological) drawing, that is, a drawing where every pair of edges intersects in at most one point. However, in combination with other restrictions simple drawings do not universally exist. For instance, k-planar graphs are those graphs that can be drawn so that every edge has at most k crossings (i.e., they admit a k-plane drawing). It is known that for k≤3 , every k-planar graph admits a k-plane simple drawing. But for k≥4 , there exist k-planar graphs that do not admit a k-plane simple drawing. Answering a question by Schaefer, we show that there exists a function Open image in new window such that every k-planar graph admits an f(k)-plane simple drawing, for all Open image in new window. Note that the function f depends on k only and is independent of the size of the graph. Furthermore, we develop an algorithm to show that every 4-planar graph admits an 8-plane simple drawing. 

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3. How to morph graphs on the torus

   Chambers, Erin Wolf ( January 2021 , Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms)

   null (Ed.)

   We present the first algorithm to morph graphs on the torus. Given two isotopic essentially 3-connected embeddings of the same graph on the Euclidean flat torus, where the edges in both drawings are geodesics, our algorithm computes a continuous deformation from one drawing to the other, such that all edges are geodesics at all times. Previously even the existence of such a morph was not known. Our algorithm runs in O(n1+ω/2) time, where ω is the matrix multiplication exponent, and the computed morph consists of O(n) parallel linear morphing steps. Existing techniques for morphing planar straight-line graphs do not immediately generalize to graphs on the torus; in particular, Cairns' original 1944 proof and its more recent improvements rely on the fact that every planar graph contains a vertex of degree at most 5. Our proof relies on a subtle geometric analysis of 6-regular triangulations of the torus. We also make heavy use of a natural extension of Tutte's spring embedding theorem to torus graphs. 

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4. Universal Geometric Graphs

   https://doi.org/10.1007/978-3-030-60440-0_14  

   Frati, Fabrizio ; Hoffmann, Michael ; Toth, Csaba D. ( October 2020 , Graph-Theoretic Concepts in Computer Science (WG 2020))

   null (Ed.)

   We introduce and study the problem of constructing geometric graphs that have few vertices and edges and that are universal for planar graphs or for some sub-class of planar graphs; a geometric graph is universal for a class H of planar graphs if it contains an embedding, i.e., a crossing-free drawing, of every graph in H . Our main result is that there exists a geometric graph with n vertices and O(nlogn) edges that is universal for n-vertex forests; this extends to the geometric setting a well-known graph-theoretic result by Chung and Graham, which states that there exists an n-vertex graph with O(nlogn) edges that contains every n-vertex forest as a subgraph. Our O(nlogn) bound on the number of edges is asymptotically optimal. We also prove that, for every h>0 , every n-vertex convex geometric graph that is universal for the class of the n-vertex outerplanar graphs has Ωh(n2−1/h) edges; this almost matches the trivial O(n2) upper bound given by the n-vertex complete convex geometric graph. Finally, we prove that there is an n-vertex convex geometric graph with n vertices and O(nlogn) edges that is universal for n-vertex caterpillars. 

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5. On Weak Flexibility in Planar Graphs

   https://doi.org/10.1007/s00373-022-02564-1  

   Lidický, Bernard ; Masařík, Tomáš ; Murphy, Kyle ; Zerbib, Shira ( December 2022 , Graphs and Combinatorics)

   Abstract Recently, Dvořák, Norin, and Postle introduced flexibility as an extension of list coloring on graphs (J Graph Theory 92(3):191–206, 2019, https://doi.org/10.1002/jgt.22447 ). In this new setting, each vertex v in some subset of V ( G ) has a request for a certain color r ( v ) in its list of colors L ( v ). The goal is to find an L coloring satisfying many, but not necessarily all, of the requests. The main studied question is whether there exists a universal constant $$\varepsilon >0$$ ε > 0 such that any graph G in some graph class $$\mathscr {C}$$ C satisfies at least $$\varepsilon$$ ε proportion of the requests. More formally, for $$k > 0$$ k > 0 the goal is to prove that for any graph $$G \in \mathscr {C}$$ G ∈ C on vertex set V , with any list assignment L of size k for each vertex, and for every $$R \subseteq V$$ R ⊆ V and a request vector $$(r(v): v\in R, ~r(v) \in L(v))$$ ( r ( v ) : v ∈ R , r ( v ) ∈ L ( v ) ) , there exists an L -coloring of G satisfying at least $$\varepsilon |R|$$ ε | R | requests. If this is true, then $$\mathscr {C}$$ C is called $$\varepsilon$$ ε - flexible for lists of size k . Choi, Clemen, Ferrara, Horn, Ma, and Masařík (Discrete Appl Math 306:20–132, 2022, https://doi.org/10.1016/j.dam.2021.09.021 ) introduced the notion of weak flexibility , where $$R = V$$ R = V . We further develop this direction by introducing a tool to handle weak flexibility. We demonstrate this new tool by showing that for every positive integer b there exists $$\varepsilon (b)>0$$ ε ( b ) > 0 so that the class of planar graphs without $$K_4, C_5 , C_6 , C_7, B_b$$ K 4 , C 5 , C 6 , C 7 , B b is weakly $$\varepsilon (b)$$ ε ( b ) -flexible for lists of size 4 (here $$K_n$$ K n , $$C_n$$ C n and $$B_n$$ B n are the complete graph, a cycle, and a book on n vertices, respectively). We also show that the class of planar graphs without $$K_4, C_5 , C_6 , C_7, B_5$$ K 4 , C 5 , C 6 , C 7 , B 5 is $$\varepsilon$$ ε -flexible for lists of size 4. The results are tight as these graph classes are not even 3-colorable. 

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