Related papers: A note on equitable colorings of forests
A graph $G$ is equitably $k$-choosable if, for every $k$-uniform list assignment $L$, $G$ is $L$-colorable and each color appears on at most $\left\lceil |V(G)|/k\right\rceil$ vertices. Equitable list-coloring was introduced by Kostochka,…
We present some partial results concerning a-T-menability of groups acting on trees. Various known results are given uniform proofs.
We give a simple proof of a recently result concerning Hardy $q$-inequalities.
This is a short addendum to a note of Beauville on the subject of the title. We prove an inequality that takes into account the constant part of the Jacobian.
A graph $G$ is list point $k$-arborable if, whenever we are given a $k$-list assignment $L(v)$ of colors for each vertex $v\in V(G)$, we can choose a color $c(v)\in L(v)$ for each vertex $v$ so that each color class induces an acyclic…
A graph on $n$ vertices is equitably $k$-colorable if it is $k$-colorable and every color is used either $\left\lfloor n/k \right\rfloor$ or $\left\lceil n/k \right\rceil$ times. Such a problem appears to be considerably harder than vertex…
The existence of a partition of the common set of the vertices of two forests into two subsets, when difference of their capacities in the neighborhood of each vertex of each forest not greater than 2 is proved, and an example, which shows…
A fair clustering instance is given a data set $A$ in which every point is assigned some color. Colors correspond to various protected attributes such as sex, ethnicity, or age. A fair clustering is an instance where membership of points in…
This note presents an application of the quasi-conform transformation in surveying.
We prove upper and lower bounds on the chromatic number of the square of the cartesian product of trees. The bounds are equal if each tree has even maximum degree.
Richard P. Stanley conjectured that finite trees can be distinguished by their chromatic symmetric functions. In this paper, we prove an analogous statement for posets: Finite rooted trees can be distinguished by their order quasisymmetric…
We present a new approach to the global fairness verification of tree-based classifiers. Given a tree-based classifier and a set of sensitive features potentially leading to discrimination, our analysis synthesizes sufficient conditions for…
This short note present a "proof" of $P\neq NP$. The "proof" with double quotation marks is to indicate that we do not know whether the proof is correct or not (We're confused because we do know in which we make the mistakes).
This paper is currently undergoing revision.
An $n$-vertex graph is equitably $k$-colorable if there is a proper coloring of its vertices such that each color is used either $\left\lfloor n/k \right\rfloor$ or $\left\lceil n/k \right\rceil$ times. While classic Vertex Coloring is…
Trees or rooted trees have been generously studied in the literature. A forest is a set of trees or rooted trees. Here we give recurrence relations between the number of some kind of rooted forest with $k$ roots and that with $k+1$ roots on…
It is shown that a natural notion of congruence permutability for quasivarieties already implies ``being a variety''. The result follows immediately from [3] and the sole aim of this note is to state it explicitly, together with a…
We define an algorithm k which takes a connected graph G on a totally ordered vertex set and returns an increasing tree R (which is not necessarily a subtree of G). We characterize the set of graphs G such that k(G)=R. Because this set has…
Given $K_n$ whose edges are coloured red and blue, and a forest $F$ of order $n$, we seek embeddings of $F$ with small imbalance, that is, difference between the numbers of red and blue edges. We show that if the $2$-colouring of the edges…
We prove several results about the complexity of the role colouring problem. A role colouring of a graph $G$ is an assignment of colours to the vertices of $G$ such that two vertices of the same colour have identical sets of colours in…