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Register a new userCounting locally flat-foldable origami configurations via 3-coloring graphs
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Origami, where two-dimensional sheets are folded into complex structures, is proving to be rich with combinatorial and geometric structure, most of which remains to be fully understood. In this paper we consider emph{flat origami}, where the sheet of material is folded into a two-dimensional object, and consider the mountain (convex) and valley (concave) creases made by such foldings, called a emph{MV assignment} of the crease pattern. We establish a method to, given a flat-foldable crease pattern $C$ under certain conditions, create a planar graph $C^*$ whose 3-colorings are in one-to-one correspondence with the locally-valid MV assignments of $C$. This reduces the general, unsolved problem of enumerating locally-valid MV assignments to the enumeration of 3-colorings of graphs.
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Rigidly and flat-foldable quadrilateral mesh origami is the class of quadrilateral mesh crease patterns with one fundamental property: the patterns can be folded from flat to fully-folded flat by a continuous one-parameter family of piecewise affine deformations that do not stretch or bend the mesh-panels. In this work, we explicitly characterize the designs and deformations of all possible rigidly and flat-foldable quadrilateral mesh origami. Our key idea is a rigidity theorem (Theorem 3.1) that characterizes compatible crease patterns surrounding a single panel and enables us to march from panel to panel to compute the pattern and its corresponding deformations explicitly. The marching procedure is computationally efficient. So we use it to formulate the inverse problem: to design a crease pattern to achieve a targeted shape along the path of its rigidly and flat-foldable motion. The initial results on the inverse problem are promising and suggest a broadly useful engineering design strategy for shape-morphing with origami.
DP-coloring is a generalization of list coloring, which was introduced by Dvov{r}{a}k and Postle [J. Combin. Theory Ser. B 129 (2018) 38--54]. Zhang [Inform. Process. Lett. 113 (9) (2013) 354--356] showed that every planar graph with neither adjacent triangles nor 5-, 6-, 9-cycles is 3-choosable. Liu et al. [Discrete Math. 342 (2019) 178--189] showed that every planar graph without 4-, 5-, 6- and 9-cycles is DP-3-colorable. In this paper, we show that every planar graph with neither adjacent triangles nor 5-, 6-, 9-cycles is DP-3-colorable, which generalizes these results. Yu et al. gave three Bordeaux-type results by showing that (i) every planar graph with the distance of triangles at least three and no 4-, 5-cycles is DP-3-colorable; (ii) every planar graph with the distance of triangles at least two and no 4-, 5-, 6-cycles is DP-3-colorable; (iii) every planar graph with the distance of triangles at least two and no 5-, 6-, 7-cycles is DP-3-colorable. We also give two Bordeaux-type results in the last section: (i) every plane graph with neither 5-, 6-, 8-cycles nor triangles at distance less than two is DP-3-colorable; (ii) every plane graph with neither 4-, 5-, 7-cycles nor triangles at distance less than two is DP-3-colorable.
We survey more recent attempts at enumerating the number of mountain-valley assignments that allow a given crease pattern to locally fold flat. In particular, we solve this problem for square twist tessellations and generalize the method used to a broader family of crease patterns. We also describe the more difficult case of the Miura-ori and a recently-discovered bijection with 3-vertex colorings of grid graphs.
For a given graph $G$, the least integer $kgeq 2$ such that for every Abelian group $mathcal{G}$ of order $k$ there exists a proper edge labeling $f:E(G)rightarrow mathcal{G}$ so that $sum_{xin N(u)}f(xu) eq sum_{xin N(v)}f(xv)$ for each edge $uvin E(G)$ is called the textit{group twin chromatic index} of $G$ and denoted by $chi_g(G)$. This graph invariant is related to a few well-known problems in the field of neighbor distinguishing graph colorings. We conjecture that $chi_g(G)leq Delta(G)+3$ for all graphs without isolated edges, where $Delta(G)$ is the maximum degree of $G$, and provide an infinite family of connected graph (trees) for which the equality holds. We prove that this conjecture is valid for all trees, and then apply this result as the base case for proving a general upper bound for all graphs $G$ without isolated edges: $chi_g(G)leq 2(Delta(G)+{rm col}(G))-5$, where ${rm col}(G)$ denotes the coloring number of $G$. This improves the best known upper bound known previously only for the case of cyclic groups $mathbb{Z}_k$.
The textit{$k$-weak-dynamic number} of a graph $G$ is the smallest number of colors we need to color the vertices of $G$ in such a way that each vertex $v$ of degree $d(v)$ sees at least $rm{min}{k,d(v)}$ colors on its neighborhood. We use reducible configurations and list coloring of graphs to prove that all planar graphs have 3-weak-dynamic number at most 6.
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