Vytautas Gruslys, Shoham Letzter
We consider the problem of minimising the number of edges that are contained in triangles, among $n$-vertex graphs with a given number of edges. We prove a conjecture of Füredi and Maleki that gives an exact formula for this minimum, for sufficiently large $n$.
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We give an asymptotic formula for the minimum number of edges contained in triangles in a graph having n vertices and e edges. Our main tool is a generalization of Zykov's symmetrization method that can be applied for several graphs simultaneously.
The vertex-edge marking game is played between two players on a graph, $G=(V,E)$, with one player marking vertices and the other marking edges. The players want to minimize/maximize, respectively, the number of marked edges incident to an unmarked vertex. The vertex-edge coloring number for $G$ is the maximum score achievable with perfect play. Brešar et al., [4], give an upper bound of $5$ for the vertex-edge coloring number for finite planar graphs. It is not known whether the bound is tight. In this paper, in response to questions in [4], we show that the vertex-edge coloring number for the…
Slitherlink on Triangular Grids
本文研究了三角网格中Slitherlink签名的性质，证明了相同签名环的对称差大小能被12整除，并完全刻画了完全偶子集。
A function $f$ is a \textit{graceful labelling} of a graph $G=(V,E)$ with $m$ edges if $f$ is an injection $f:V\mapsto \{0,1,2,\dots,m\}$ such that each edge $uv \in E$ is assigned the label $|f(u)-f(v)|$, and no two edge labels are the same. If a graph G has a graceful labelling, we say that $G$ itself is graceful. In this paper, we prove any Dutch windmill with three pendant triangles is (near) graceful, which settles Rosa's conjecture for a new family of triangular cacti.
A regular graph is co-edge regular if there exists a constant $μ$ such that any two distinct and non-adjacent vertices have exactly $μ$ common neighbors. In this paper, we show that for integers $s\ge 2$ and $n$ large enough, any co-edge-regular graph which is cospectral with the $s$-clique extension of the triangular graph $T((n)$ is exactly the $s$-clique extension of the triangular graph $T(n)$.
Given a graph G, its triangular line graph is the graph T(G) with vertex set consisting of the edges of G and adjacencies between edges that are incident in G as well as being within a common triangle. Graphs with a representation as the triangular line graph of some graph G are triangular line graphs, which have been studied under many names including anti-Gallai graphs, 2-in-3 graphs, and link graphs. While closely related to line graphs, triangular line graphs have been difficult to understand and characterize. Van Bang Le asked if recognizing triangular line graphs has an efficient algorit…
Resistance values under transformations in regular triangular grids
本文拓展了三角网格电阻距离的相关研究，通过等效电路逐行约简分析非渐近电阻行为，提出并证明了电阻值与1大小关系猜想的特例，还发现了约简后对角线上的重复电阻模式，同时优化了相关符号与证明方法。
Balanced Spanning Trees for Triangular Strip Lattices
本文针对三角条带格，通过递推关系和渐近分析，研究了平衡生成树的计数及其比例，并给出了均匀随机选取时平衡生成树的极限概率。
The Turán number of a graph $H$, denoted by $\text{ex}(n, H)$, is the maximum number of edges in an $n$-vertex graph that does not have $H$ as a subgraph. Let $TP_k$ be the triangular pyramid of $k$-layers. In this paper, we determine that $\text{ex}(n,TP_3)= \frac{1}{4}n^2+n+o(n)$ and pose a conjecture for $\text{ex}(n,TP_4)$.
A signed graph is a simple graph with two types of edges. Switching a vertex $v$ of a signed graph corresponds to changing the type of each edge incident to $v$. A homomorphism from a signed graph $G$ to another signed graph $H$ is a mapping $φ: V(G) \rightarrow V(H)$ such that, after switching any number of the vertices of $G$, $φ$ maps every edge of $G$ to an edge of the same type in $H$. The chromatic number $χ_s(G)$ of a signed graph $G$ is the order of a smallest signed graph $H$ such that there is a homomorphism from $G$ to $H$. We show that the chromatic number of signed triangular grid…
A trivalent diagram is a connected, two-colored bipartite graph (parallel edges allowed but not loops) such that every black vertex is of degree 1 or 3 and every white vertex is of degree 1 or 2, with a cyclic order imposed on every set of edges incident to to a same vertex. A rooted trivalent diagram is a trivalent diagram with a distinguished edge, its root. We shall describe and analyze an algorithm giving an exhaustive list of rooted trivalent diagrams of a given size (number of edges), the list being non-redundant in that no two diagrams of the list are isomorphic. The algorithm will be s…
Degree sequences of triangular multigraphs
本文研究三角多重图的度序列问题，将Egan和Nikolayevsky关于三角简单图的猜想推广到多重图情形，并证明了每一项至少为4的可图序列均可实现为三角多重图的度序列。
Two problems on booksize and triangular edges in Nosal graphs
本文研究Nosal图中书尺寸和三角形边的问题，证明了无孤立顶点且非完全二分图的Nosal图的书尺寸和三角形边数下限。
Edge-decomposition into Two Triangular Forests is NP-complete
研究将图边分解为两个三角森林问题的复杂度，通过考虑满足特定条件的最简单图类三角森林，证明该边分解问题是NP完全的。
Limiting Behavior of Resistances in Triangular Graphs
本文通过计算研究三角图电阻的行约简算法，提出相关极限行为猜想并证明一类图的边比关系与图对称性的关联，还引入了等效电阻恒等式的机械验证方法。
Finding necessary conditions for the geometry of flexible polyhedra is a classical problem that has received attention also in recent times. For flexible polyhedra with triangular faces, we showed in a previous work the existence of cycles with a sign assignment for their edges, such that the signed sum of the edge lengths along the cycle is zero. In this work, we extend this result to flexible non-triangular polyhedra.
Let $G=(V(G),E(G))$ be a planar digraph embedded in the plane in which all inner faces are equilateral triangles (with three edges in each), and let the union $\Rscr$ of these faces forms a convex polygon. The question is: given a function $σ$ on the boundary edges of $G$, does there exist a concave function $f$ on $\Rscr$ which is affinely linear within each bounded face and satisfies $f(v)-f(u)=σ(e)$ for each boundary edge $e=(u,v)$? The functions $σ$ admitting such an $f$ form a polyhedral cone $C$, and when the region $\Rscr$ is a triangle, $C$ turns out to be exactly the cone of boundary …
We show that a matchstick graph with $n$ vertices has no more than $3n-c\sqrt{n-1/4}$ edges, where $c=\frac12(\sqrt{12} + \sqrt{2π\sqrt{3}})$. The main tools in the proof are the Euler formula, the isoperimetric inequality, and an upper bound for the number of edges in terms of $n$ and the number of non-triangular faces. We also find a sharp upper bound for the number of triangular faces in a matchstick graph.
Side-to-side Tiling of the Sphere by Congruent Curvilinear Triangles
该研究分类了球面侧向等边曲边三角形镶嵌，发现其本质是直线三角形镶嵌的变形。
The Turán number of the triangular pyramid of 4-layers
本文证实了四层三棱锥的Turán数为1/4n²+Θ(n^{4/3})
The problem of calculating exact lower bounds for the number of $k$-faces of $d$-polytopes with $n$ vertices, for each value of $k$, and characterising the minimisers, has recently been solved for $n\le2d$. We establish the corresponding result for $n=2d+1$; the nature of the lower bounds and the minimising polytopes are quite different in this case. As a byproduct, we also characterise all $d$-polytopes with $d+3$ vertices, and only one or two edges more than the minimum.
An edge tessellation is a tiling of the plane generated by reflecting a polygon in its edges. We prove that a polygon generating an edge tessellation is one the following eight types: a rectangle; an equilateral, 60-right, isosceles right, or 120-isosceles triangle; a 120-rhombus; a 60-90-120 kite; or a regular hexagon. A stamp folding puzzle is a paper folding problem constrained to the perforations on a sheet of postage stamps. We establish the following conjecture due to G. Frederickson: "Although triangular stamps have come in a variety of different triangular shapes, only three shapes see…
For a $d$-dimensional polytope with $v$ vertices, $d+1\le v\le2d$, we calculate precisely the minimum possible number of $m$-dimensional faces, when $m=1$ or $m\ge0.62d$. This confirms a conjecture of Grünbaum, for these values of $m$. For $v=2d+1$, we solve the same problem when $m=1$ or $d-2$; the solution was already known for $m= d-1$. In all these cases, we give a characterisation of the minimising polytopes. We also show that there are many gaps in the possible number of $m$-faces: for example, there is no polytope with 80 edges in dimension 10, and a polytope with 407 edges can have dim…
Improved bounds for the triangle case of Aharoni's rainbow generalization of the Caccetta-Häggkvist conjecture
本文改进了Aharoni关于Caccetta-Häggkvist猜想彩虹推广的三角形式界，证明了(1.1077,1/3)、(1,0.3988)及(1.3481,1/4)等参数对是三角的。
Tilings of the sphere by congruent regular triangles and congruent rhombi
本文对使用全等正三角形和全等菱形对球面进行的所有边对边平铺进行了完整分类，包括参数化族和26个零散原型集，并提供了参数化平铺的模空间及所有几何数据。
Uniform integer-valued Lipschitz functions on a domain of size $N$ of the triangular lattice are shown to have variations of order $\sqrt{\log N}$. The level lines of such functions form a loop $O(2)$ model on the edges of the hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for the loop O(2) model is constructed as a thermodynamic limit and is shown to be unique. It contains only finite loops and has properties indicative of scale-invariance: macroscopic loops appearing at every scale. The existence of the infinite-volume measure carries over to height functions pinned…
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