Related papers: Fixed-point cycles and EFX allocations

We consider the Cartesian product X of n finite intervals of integers and a map F from X to itself. As main result, we establish an upper bound on the number of fixed points for F which only depends on X and on the topology of the positive…

Very recently, Alon and Frankl initiated the study of the maximum number of edges in $n$-vertex $F$-free graphs with matching number at most $s$. For fixed $F$ and $s$, we determine this number apart from a constant additive term. We also…

The maximization for the independence systems defined on graphs is a generalization of combinatorial optimization problems such as the maximum $b$-matching, the unweighted MAX-SAT, the matchoid, and the maximum timed matching problems. In…

We introduce the notion of balance for directed graphs: a weighted directed graph is $\alpha$-balanced if for every cut $S \subseteq V$, the total weight of edges going from $S$ to $V\setminus S$ is within factor $\alpha$ of the total…

For a simple graph $F$, let $\mathrm{EX}(n, F)$ and $\mathrm{EX_{sp}}(n,F)$ be the set of graphs with the maximum number of edges and the set of graphs with the maximum spectral radius in an $n$-vertex graph without any copy of the graph…

For a finite point set $E\subset \mathbb{R}^d$ and a connected graph $G$ on $k+1$ vertices, we define a $G$-framework to be a collection of $k + 1$ points in E such that the distance between a pair of points is specified if the…

We extend a recent argument of Kahn, Narayanan and Park (Proceedings of the AMS, to appear) about the threshold for the appearance of the square of a Hamilton cycle to other spanning structures. In particular, for any spanning graph, we…

Popielarz, Sahasrabudhe and Snyder in 2018 proved that maximal $K_{r+1}$-free graphs with $(1-\frac{1}{r})\frac{n^2}{2}-o(n^{\frac{r+1}{r}})$ edges contain a complete $r$-partite subgraph on $n-o(n)$ vertices. This was very recently…

This paper considers synchronous discrete-time dynamical systems on graphs based on the threshold model. It is well known that after a finite number of rounds these systems either reach a fixed point or enter a 2-cycle. The problem of…

A good edge-labeling of a graph [Ara\'ujo, Cohen, Giroire, Havet, Discrete Appl. Math., forthcoming] is an assignment of numbers to the edges such that for no pair of vertices, there exist two non-decreasing paths. In this paper, we study…

We study the fair allocation of indivisible goods under cardinality constraints, where each agent must receive a bundle of fixed size. This models practical scenarios, such as assigning shifts or forming equally sized teams. Recently,…

Alon, Krivelevich, and Sudakov conjectured in 1999 that for every finite graph $F$, there exists a quantity $c(F)$ such that $\chi(G) \leq (c(F) + o(1)) \Delta / \log\Delta$ whenever $G$ is an $F$-free graph of maximum degree $\Delta$. The…

We show, in particular, that a multivalued map $f$ from a closed subspace $X$ of $\mathbb R^n$ to ${\rm exp}_k(\mathbb R^n)$ has a point of period exactly $M$ if and only if its continuous extension $\tilde f: \beta X\to {\rm exp}_k(\beta…

We describe a way of assigning labels to the vertices of any undirected graph on up to $n$ vertices, each composed of $n/2+O(1)$ bits, such that given the labels of two vertices, and no other information regarding the graph, it is possible…

Machine learning and data mining algorithms have been increasingly used recently to support decision-making systems in many areas of high societal importance such as healthcare, education, or security. While being very efficient in their…

We study the \emph{{interval completion}} problem, which asks for the insertion of a set of at most $k$ edges to make a graph of $n$ vertices into an interval graph. We focus on chordal graphs with no small obstructions, where every…

We study two extremal problems about subgraphs excluding a family $\F$ of graphs. i) Among all graphs with $m$ edges, what is the smallest size $f(m,\F)$ of a largest $\F$--free subgraph? ii) Among all graphs with minimum degree $\delta$…

For an integer $q\ge 2$ and a graph $F$ with $q\mid e(F)$, let $R(F,\Z_q)$ be the least integer $n$ such that every edge-labeling $w\colon E(K_n)\to \Z_q$ contains a copy of $F$ whose edge-label sum is zero in $\Z_q$. Write $C_{qk}$ for the…

We consider an independently identically distributed random dynamical system generated by finitely many, non-uniformly expanding Markov interval maps with a finite number of branches. Assuming a topologically mixing condition and the…

This paper explores the conditions for determining fixed nodes in structured networks, specifically focusing on directed acyclic graphs (DAGs). We introduce several necessary and sufficient conditions for determining fixed nodes in…