# Tagged Questions

The study of maximal or minimal graphs satisfying certain properties.

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### Prove or dis-prove that it always holds or not $\lambda(G) \leq \chi(G)$ [on hold]

I want to prove that this inequality holds or not? The inequality is $\lambda(G) \leq \chi(G)$ where $\lambda(G)$ is the minimum number of edges whose deletion from a graph $G$ disconnects $G$, ...
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### Any undirected graph on 9 vertices with minimum degree at least 5 contains a subgraph $K_4$?

Let $G$ be simple undirected graph with degree of every vertices is at least 5. Prove or disprove that $G$ contains subgraph $K_4$. I came up with this question when I were trying to find Ramsey ...
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### How to find List Chromatic Number of planar graphs [on hold]

I want to know how we can find the list chromatic number of planar graphs, Suppose we have graph $G= K_{3}$. Then its chromatic number is $3$, but what is the list chromatic number of $K_{3}$? ...
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### Does the complete graph contain the maximum number of simple cycles?

Let $\mathcal{G}(n,m)$ be the set of connected, simple graphs with $n$ vertices and $m$ edges. For any graph $G$ we denote its number of simple cycles with $\mu(G)$ and and for any finite family of ...
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### Show a function is bounded using Ramsey's Theorem

Suppose we have so,e bounded functions $g_1, \dots, g_k$, a function $f$, and constants $\epsilon, \delta$ such that whenever $f(x)-f(y) > \epsilon$ then $\max_i g_i(x) - g_i(y) > \delta$. I ...
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### Conditional extremes, solving $xa+yb < (x^p+y^p)^{\frac{1}{p}}(x^q+y^q)^{\frac{1}{q}}$ if.

Conditional extremes, solving $$xa+yb \leq (x^p+y^p)^{\frac{1}{p}}(x^q+y^q)^{\frac{1}{q}}$$ using lagrange multipliers.. If $\frac{1}{q}+\frac{1}{p}=1$ and $p,q>1$. This reminds me of Holders ...
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### Relative Extrema - First-derivative test of : $f(x)=x^5-5x^3-20x-2$

Find the relative extrema of the function by applying the first-derivative test: $$f(x)=x^5-5x^3-20x-2$$ So I found the $f'(x)$ $$f'(x) = 5x^4-15x^2-20$$ Now, I'm trying to find the critical ...
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### n-critical graph with order n+2

Problem: Let G be an $n$-critical graph of order $n+2$. Show that $\overline{G}$ consists of $C_5$ and some isolated vertices. What I've managed to do: Not much, since I don't have many tools at my ...
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### 21 points on circumference of a circle must have at least 100 pairs separated by 120+ degrees.

Prove that at least 100 of the arcs determined by the pairs of these points subtend an angle not exceeding 120 degrees at the center. How do I prove this? Induction? Help please. Thanks.
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### Probability spaces over graphs: which area has focus on them?

Suppose a simple graph $G$. Now consider probability space $G(v;p)$ where $0\leq p\leq 1$ and $v$ vertices. I want to calculate globally-determined properties of $G(v;p)$ such as connectivity and ...
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### Conditional Extreme. Find a point in $\mathbb{R^2}$ that has the smallest sum of squared distances from the lines $x=0,y=0, x-y+1=0.$

I can find the main function, but I do not know the condition, to set up the Lagrange equation. Can anyone see, what condition the point has to satisfy here?(So as to apply the Lagrange multiplier ...
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### With the given point $M(a,b,c)$ in $\mathbb R^3$, find the tetrahedron with the smallest volume that is formed with a plane that..

With the given point $M(a,b,c)$ in $\mathbb R^3$, find the tetrahedron with the smallest volume that is formed with a plane that contains $M$ and who's points are the intersections of that plane with ...
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### Finding the point on the parabola closest to a point

Find the point on the parabola $2x - 2y^2 = 7$ which is closest to the point $(4,16)$. I've tried the distance $D$ between the point $(x,y)$ and $(4,16)$, then the problem is simplified by ...
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### Graph theory and tree company

I appreciate anyone who answer this question and I anyone who design appropriate graph.
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### Every simple planar graph with $\delta\geq 3$ has an adjacent pair with $deg(u)+deg(v)\leq 13$

Claim: Every simple planar graph with minimum degree at least three has an edge $uv$ such that $deg(u) + deg(v)\leq 13$. Furthermore, there exists an example showing that 13 cannot be replaced by 12. ...
Let $G$ be a graph with $a\choose 2$ edges (and an arbitrary number of vertices). Is it true that it has at most $a\choose 3$ triangles? Context: this continues the question Number of triangles in a ...