Lobsters with an almost perfect matching are graceful

Let $T$ be a lobster with a matching that covers all but one vertex. We show that in this case, $T$ is graceful.Comment: 4 page

In Pursuit of the Ringel-Kotzig Conjecture: Uniform K-Distant Trees are Graceful

Graph labeling has been an active area of research since 1967, when Rosa introduced the concept. Arguably, the biggest open conjecture in the field is referred to as the Ringel-Kotzig conjecture, which states that all trees admit a graceful labeling. In this talk, we will give a bit of background on the problem, as well as present our own results. Namely, that a certain infinite class of trees (called uniform k-distant trees) admits a graceful labeling

Simulation of Large Scale Computational Ecosystems with Alchemist: A Tutorial

Many interesting systems in several disciplines can be modeled as networks of nodes that can store and exchange data: pervasive systems, edge computing scenarios, and even biological and bio-inspired systems. These systems feature inherent complexity, and often simulation is the preferred (and sometimes the only) way of investigating their behavior; this is true both in the design phase and in the verification and testing phase. In this tutorial paper, we provide a guide to the simulation of such systems by leveraging Alchemist, an existing research tool used in several works in the literature. We introduce its meta-model and its extensible architecture; we discuss reference examples of increasing complexity; and we finally show how to configure the tool to automatically execute multiple repetitions of simulations with different controlled variables, achieving reliable and reproducible results

Two Rosa-type Labelings of Uniform k-distant Trees and a New Class of Trees

A k-distant tree consists of a main path, called the spine, such that each vertex on the spine is joined by an edge to an end-vertex of at most one path on at most k vertices. Those paths, along with the edge joining them to the spine, are called tails. When every vertex on the spine has exactly one incident tail of length k we call the tree a uniform k-distant tree. We show that every uniform k-distant tree admits both a graceful- and an α-labeling. For a graph G and a positive integer a, define appa(G) to be the graph obtained from appending a leaves to each leaf in G. When G is a uniform k-distant tree, we show that appa(G) admits both a graceful- and an α-labeling

Some Topics of Special Interest in Graph Theory

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Discussion On Some Important Topics in Graph Theory

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Some Investigations in the Theory of Graphs

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