## co.combinatorics – Sufficient criterion for unit distance graphs

There are many necessary criteria for a graph to be a unit distance graph. For example, it must not have $$K_4$$ as a subgraph etc. Can we find some sufficient criterion for a graph to be a unit distance graph?
This may be useful to prove/disprove that some well-known graphs are unit-distance graphs, say Petersen graph, odd graphs etc.

## co.combinatorics – Is the number of words finite, when you don’t know how to count?

This question is inspired by this one:

Can you do math without knowing how to count?

Let $$M_2$$ be the set of words constructed by concatenation of the letters $$a_1$$ and $$a_2$$, with :

(*) : for any $$x$$ word of $$M_2$$ $$xx = x$$.

Is it true $$card(M_2)=card(mathbb N)$$?

If not, is it true $$exists n in mathbb N, card(M_n)=card(mathbb N)$$?

The condition (*) comes from the hypothesis that we assume that we do not know how to count.

## co.combinatorics – Coloring finite subsets of a fixed size with a single modular functuion

Let $$k$$ and $$N$$ be positive integers so that $$k|N$$. Let $$M=(k/N){Nchoose k}$$. A function $$f:(N)^krightarrow M$$ is a coloring function if $$f(s_1) = f(s_2)$$ implies that $$s_1=s_2$$ or $$s_1 cap s_2 = emptyset$$. Coloring functions exist for all $$k$$ and $$N$$ by Baranyai’s theorem. When $$k=2$$, $$M=N-1$$ and it is widely known and easily shown that there is a function $$f(i,j)$$, such that for every even $$N$$, $$f(s)=f(i,j)bmod (N-1)$$ is a coloring function for 2 and N where $$s =langle i, j rangle$$, $$0 leq i < j < N$$. My question is whether there can exist for a $$k>2$$ an $$f(i_0,i_1, ldots i_{k-1})$$ so that $$f(langle i_0,i_1, ldots i_{k-1}rangle) = f(i_0,i_1, ldots i_{k-1})bmod M$$ is a coloring function for all $$N$$ that are multiples of $$k$$. Probably not but I’d have no idea how to prove it.

## co.combinatorics – Number of distinct pairs which doesn’t share difference

Inspired by This question by Vidyarthi I tried to find the value of $$T(2m)$$ where, $$T(2m)$$ is the number of sets of distinct pairings (so, the sets have $$m$$ elements) of the numbers $$1,2,3….,2m$$ such that for no two pairs of a set $$(a,b)$$ and $$(c,d)$$ $$|a-b|=|c-d|$$. But I couldn’t find any reasonable way to solve it. Any answers and comments are welcome.

$$T(2m)$$ is odd. Because, for a set $$(a_1,a_2);(a_3,a_4);….;(a_{2m-1},a_{2m})$$ we can swap $$a_i$$ by $$2m+1-a_i$$ for all $$i$$. So there is a dual set for each set of pairs except for $${(1,2m);(2,2m-1);…;(m-1,m+1)}$$ whose swapped-pair is this itself.

Using $$T(2m)$$ we may show directly the referenced question: Let, we have a set of differences (in modulo $$k+1$$) which has no such solution pairs (as asked in the referenced question) be $$l_1,l_2…,l_m$$ then in this question all sets of differences $$(l_1,k+1-l_1);(l_2,k+1-l_2);…;(l_m,k+1-l_m)$$ are prohibited (taking one from each bracket/ due to the modulo argument). Hence, total $$2^m$$ are prevented. If, we can show that $$T(2m)>binom{2m-1}{m}-2^m=frac{1}{2}binom{2m}{m}-2^m$$ then, the referenced question is proved.

e.g for $$2m=6$$, $$T(6)=5$$ and $$binom{6-1}{3}-2^3=2$$. Also,for $$2m=8, T(8)>binom{7}{4}-2^4=19$$

## co.combinatorics – What are the coloops of a hypergraphic matroid?

For a hypergraph $$H=(V,E)$$ call $$Fsubseteq E$$ a hyperforest in $$H$$ iff there is a forest graph $$G$$ and a bijection $$smallphi:E(G)to F$$ satisfying $$smallforall ein E(G)(esubseteq phi(e))$$ or equivalently $$smallforall Isubseteq F(|I|+1leq |cup_{Sin I}S|)$$, further we call the matroid $$M(H)=(E,mathcal{I})$$ such that $$mathcal{I}={Isubseteq E:Itext{ is a hyperforest in }H}$$ the hypergraphic matroid of $$H$$ so in particular if $$H$$ is an undirected graph then this coincides with the standard definition described here while analogously if $$mathcal{E}in E$$ then $$M(Hsetminusmathcal{E})=M(H)setminusmathcal{E}$$.

Now with all of that said, my question is given any hypergraph $$H$$ what are the coloops of $$M(H)$$?

I suspect they relate to the line graph $$smallmathcal{L}(H)=(E(H),{{X,Y}subseteq E(H):Xneq Ytext{ and }Xcap Yneqemptyset})$$.

## co.combinatorics – Fastest algorithm to construct a proper edge \$(Delta(G)+1)\$-coloring of a simple graph

A proper edge coloring is a coloring of the edges of a graph so that adjacent edges receive distinct colors. Vizing’s theorem states that every simple graph $$G$$ has a proper edge coloring using at most maximum degree plus one colors. In (1), the authors showed that there is an $$O(mn)$$-time algorithm to construct a proper edge $$(Delta(G)+1)$$-coloring of a simple graph whth $$m$$ edges and $$n$$ vertices. I wonder whether there is a faster algorithm to construct such an edge coloring.

(1) J. Misra, and D. Gries. A constructive proof of Vizing’s theorem.Inform. Process. Lett.41(3)131–133 (1992)

## co.combinatorics – A ratio of two probabilities

Ley $$t:=eta$$. Then
$$f(t)=frac{P(Gge K)}{P(B ge K)},$$
where $$G$$ is a random variable with the binomial distribution with parameters $$N,q_Gt$$ and $$B$$ is a random variable with the binomial distribution with parameters $$N,q_Bt$$; here we must assume that $$q_B>0$$ and $$tin(0,1/g_G)$$, so that $$q_Gt$$ and $$q_Bt$$ are in the interval $$(0,1)$$.

The random variables $$G$$ and $$B$$ have a monotone likelihood ratio (MLR): for each $$xin{0,dots,N}$$,
$$frac{P(G=x)}{P(B=x)}=CBig(frac{1-q_Gt}{1-q_Bt}Big)^{N-x},$$
which is decreasing in $$tin(0,1/g_G)$$; here, $$C$$ is a positive real number which does not depend on $$t$$.

It is well known that the MLR implies the MTR, the monotone tail ratio. Thus, the desired result follows.

## co.combinatorics – Find a collection of values of polynomial

Given a polynomial $$f(x)in mathbb C(x)$$ where $$deg f(x)=n-1.$$ Assume that we need to find a collection of values of this polynomial corresponding to the following set of $$x$$-values: $${ e^{ik} }$$ where $$k= 0, …, n-1,$$ and $$i$$ denotes the imaginary unit.

There is an algorithm that can solve this problem for any collection of $$x$$-values using FFT in $$O(nlog^2(n)).$$
On the other hand, if all $$x$$-values are $$n$$ – roots of unity, then FFT can solve the problem in $$O(nlog n).$$

Question: Is it possible to solve the above problem in $$O(nlog n)$$ as well?

## co.combinatorics – Number of linear inequalities describing a polyhedron with prescribed number of vertices

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## co.combinatorics – Packing equal-size disks in a unit disk

Inspired by the delicious buns and Siu Mai in bamboo steamers I saw tonight in a food show about Cantonese Dim Sum, here is a natural question. It probably has been well studied in the literature, but I cannot find related reference.

Given a real number $$r le 1$$, let $$f(r)$$ be the maximum number of radius-$$r$$ disks that can be packed into a unit disk. For example, $$f(1)=1$$ for $$r in (1/2, 1)$$, $$f(r)=2$$ for $$r in (2sqrt{3}-3, 1/2)$$, etc.

Question: Is it true that $${f(r): r in (0, 1)}=mathbb{N}$$?