We study edge-sum distinguishing labeling, a type of labeling recently introduced by Z. Tuza (2017) in context games. An ESD an $n$-vertex graph $G$ is injective mapping integers $1$ to $l$ its vertices such that for every edge, the sum on endpoints unique. If $ l$ equals $n$, we speak about canonical labeling. focus primarily structural properties this and show several classes graphs if they h...

In a letter to his friend Burgundio (Sidon. epist. 9.14), who was about write laus Caesaris, Sidonius Apollinaris drew up short list of the works written on Caesar that had face: scripta Patavinis, opera Svetonii, ephemeris Balbi, and finally Iuventii Martialis historia. If mention Livy, Suetonius, Balbus is not surprising, name Iuventius (or Vivencius in Parisinus Latinus 9551) raises some pro...

The study of substructures in random objects has a long history, beginning with Erdős and Renyi's work on subgraphs graphs. We the existence certain subsets vector spaces over finite fields. First we provide general framework which can be applied to establish coarse threshold results prove limiting Poisson distribution at scale. To illustrate our apply $k$-term arithmetic progressions, sums, ri...