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 – 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 ${f(r): r in (0, 1)}=mathbb{N}$?