Given an edge-coloured graph, we say that a subgraph is rainbow if all of its edges have different colours. Let $\operatorname{ex}(n,H,$rainbow-$F)$ denote the maximal number copies $H$ properly graph on $n$ vertices can contain it has no isomorphic to $F$. We determine order magnitude $\operatorname{ex}(n,C_s,$rainbow-$C_t)$ for $s,t$ with $s\not =3$. In particular, answer question Gerbner, M\...

The graph-search algorithms developed between 60s and 80s were widely used in many fields, from robotics to video games. The A* algorithm shall be mentioned between some of the most important solutions explicitly oriented to motion-robotics, improving the logic of graph search with heuristic principles inside the loop. Nevertheless, one of the most important drawbacks of the A* algorithm reside...

We present a construction called layered wheel. Layered wheels are graphs of arbitrarily large treewidth and girth. They might be an outcome for possible theorem characterizing with in terms their induced subgraphs (while such characterization is well-understood minors). also provide examples rankwidth well-studied classes, as (theta, triangle)-free even-hole-free no K 4 (where hole chordless c...

We study the relationship between unit-distance representations and the Lovász theta number of graphs, originally established by Lovász. We derive and prove min-max theorems. This framework allows us to derive a weighted version of the hypersphere number of a graph and a related min-max theorem. Then, we connect to sandwich theorems via graph homomorphisms. We present and study a generalization...

Introduction: Contrary to Diagnostic and Statistical Manual of Mental Disorders (DSM-5), fifth edition, some studies indicate that ADHD-inattentive presentation (ADHD-I) is a distinct diagnostic disorder and not an ADHD presentation.  Methods: In this study, 12 ADHD-combined presentation (ADHD-C), 10 ADHD-I, and 13 controls were enrolled and their resting state EEG recorded. Following thi...

We show that there is a polynomial time algorithm that, given three vertices of a graph, tests whether there is an induced subgraph that is a tree, containing the three vertices. (Indeed, there is an explicit construction of the cases when there is no such tree.) As a consequence, we show that there is a polynomial time algorithm to test whether a graph contains a “theta” as an induced subgraph...

Lovász and Schrijver [LS91] devised a lift-and-project method that produces a sequence of convex relaxations for the problem of finding in a graph an independent set(or a clique) of maximum size. Each relaxation in the sequence is tighter than the one before it, while the first relaxation is already at least as strong as the Lovász theta function [Lov79]. We show that on a random graph Gn,1/2, ...

This paper introduces a new proximity graph, called the k-Semi-Yao graph (k-SYG), on a set P of points in R, which is a supergraph of the k-nearest neighbor graph (k-NNG) of P . We provide a kinetic data structure (KDS) to maintain the k-SYG on moving points, where the trajectory of each point is a polynomial function whose degree is bounded by some constant. Our technique gives the first KDS f...

In online list coloring (introduced by Zhu and by Schauz in 2009), on each round the set of vertices having a particular color in their lists is revealed, and the coloring algorithm chooses an independent subset of this set to receive that color. For a graph G and a function f : V (G) → N, the graph is f -paintable if there is an algorithm to produce a proper coloring when each vertex v is allo...