62 problems

Li–Peng conjecture. For every $m$-edge Nosal graph $G$,

Conlon–Fox–Sudakov conjecture. If $b(G)\le b$ and $G$ is not the balanced complete bipartite graph, then $G$ has at least

For a simple drawing of $K_n$, let $\triangle(n)$ denote the minimum number of empty triangles among all such drawings. An empty triangle is a triangle induced by three vertices su…

Let $G$ be a graph with $m$ edges, and let $\lambda(G)$ denote its spectral radius. An edge is triangular if it lies in a triangle. Li--Feng--Peng's conjecture. If … then $G$ conta…

Li–Feng–Peng conjecture. If

Zhai–Lin–Shu conjecture. Every $m$-edge Nosal graph $G$ should have large booksize.

Triangle eigenvalue bound. The eigenvalue satisfies

Limit-existence conjecture. In the statement above, the limits defining the ratios of logarithmic cardinalities for all three graph families exist; in particular, each correspondin…

Li–Liu–Zhang conjecture. Every $m$-edge Nosal graph has more than $\sqrt{m}$ triangular edges.

Let $\triangle$ be a triangle, let $\lambda_1(\triangle)$ be its first Dirichlet eigenvalue, and let … be its Cheeger constant. Parini's triangle conjecture. For any triangle…

Let $\triangle$ be a triangle, with area $|\triangle|$, perimeter $|\partial\triangle|$, and first Dirichlet eigenvalue $\lambda_1(\triangle)$. Antunes–Freitas conjecture. There ex…

Let $\triangle$ be a triangle, with area $|\triangle|$, perimeter $|\partial\triangle|$, and first Dirichlet eigenvalue $\lambda_1(\triangle)$. Siudeja's conjecture. For any triang…

Laugesen–Siudeja's conjecture. The functional $\mathcal{F}(\triangle)$ is minimized uniquely by the equilateral triangle, and its minimum value is $7\sqrt{3}\pi^2/12$. This is a sh…

Let $n$ and $k$ be integers with $n\geq 3k$, let $kC_3$ be the vertex-disjoint union of $k$ triangles, and let $G_1(n,k),\ldots,G_4(n,k)$ be the four explicitly defined edge-colore…

For positive integers $n$ and $k$ with $n\geq 3k$, let $kC_3$ denote the vertex-disjoint union of $k$ triangles, and let $ar(n,kC_3)$ be the maximum number of colors in an edge-col…

Let $n$ and $k$ be positive integers with $n\geq 3k$. For an edge-colored graph $G$ of order $n$, let $\delta^c(G)$ denote its minimum color degree, and call a triangle rainbow whe…

Let $T$ be an arbitrary triangle. For each admissible choice of boundary sides, let $\lambda_{1}^{S}$, $\lambda_{1}^{M}$, $\lambda_{1}^{L}$, and $\lambda_{1}^{MS}$ denote the corre…

Let $T$ be a bounded triangle in the Euclidean plane, and let the Dirichlet eigenvalues of its Laplacian, listed with multiplicity, be … An equilateral triangle is a triangle whose…

Let $k\geq 2$ and let $n$ be sufficiently large. For $1\geq\alpha,\beta,\gamma\geq 0$, an $(k,n)$-cyclic triple is a triple satisfying the six cyclic density inequalities stated in…

Let $G$ be an $n$-vertex graph with $m$ edges. Define … Füredi–Maleki's numerical conjecture. The graph $G$ has at least $g(n,m)$ triangular edges. This is the numerical extremal f…

Let $m>n^2/4$ and let $G$ be an $n$-vertex graph with $m$ edges that minimizes the number of triangular edges. For integers $a,b,c$, let $G(a,b,c)$ be the graph consisting of a cli…

Triangle fail-point conjecture. The fail point lies on the longest side and is always closer to the foot of the altitude onto that side than to the midpoint of the side.

Let $N\in\mathbb N$ and let $\theta\in(0,\pi)$. Consider triangles of area $1$ having a fixed interior angle $\theta$, and let $\rho_N$ denote their Bergman $N$-polynomial content…

Cambie–de Verclos–Kang conjecture. For every odd integer $n$ and even integer $k$ with

Let $S$ be a point set with $3n$ elements in the plane in general position. An intersecting edge-disjoint triangle family is a family of edge-disjoint triangles such that every two…