30 problems

Triangle eigenvalue bound. The eigenvalue satisfies

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

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…

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…

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

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 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 $T$ 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 a graph with $m$ edges, let $\lambda(G)$ denote its spectral radius, and let $t(G)$ denote the number of triangular edges. Spectral triangular-edge counting conjecture.…

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…

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…

Let $kC_3$ denote the union of $k$ vertex-disjoint copies of the triangle $C_3$. For a positive integer $n$, let $ar(n,kC_3)$ be the maximum number of colors in an edge-coloring of…

Let $T_1$ and $T_2$ be triangles, and write $\lambda_i(T_j)$ for their $i$th eigenvalues of the Dirichlet Laplacian. Antunes–Freitas conjecture. If … then $T_1$ and $T_2$ are ident…

Let $G$ be a graph on $n$ vertices with maximum degree $d$. Write … Here $q$ and $r$ are the quotient and remainder in the division of $n$ by $d+1$. Gan–Loh–Sudakov's conjecture. T…

Let $t$ be an integer with $t\geq 4$. For an $n$-vertex $K_4$-saturated graph, let $\operatorname{sat}_t(n,K_3,K_4)$ denote the minimum number of triangles among graphs with minimu…

Let $G$ be a graph on $n$ vertices, and let $b(G)$ be its book number. For nonnegative integers $b$ and $n$ with $b\leq n/4$, let $S_{b,n}$ be the graph obtained by blowing up the…

Many-colour minimum colour-class conjecture. If

Many-colour edge-density conjecture. If

Let $g_3(n,e)$ be the minimum number of triangles in an $(n,e)$-graph. Let $\mathcal{H}_0^*(n,e)$ and $\mathcal{H}_2^*(n,e)$ denote the subclasses of $\mathcal{H}_0(n,e)$ and…

Erdős's conjecture. Any set of seven points in the plane determines at least four distinct triangles; consequently,

Let $G$ be a graph with $n$ vertices, $t$ triangles, and local triangle bound $y$. Here, the local triangle bound $y$ is an upper bound on the number of triangles containing any gi…