On the Graceful Game

A graceful labeling of a graph $G$ with $m$ edges consists of labeling the vertices of $G$ with distinct integers from $0$ to $m$ such that, when each edge is assigned as induced label the absolute difference of the labels of its endpoints, all induced edge labels are distinct. Rosa established two well known conjectures: all trees are graceful (1966) and all triangular cacti are graceful (1988). In order to contribute to both conjectures we study graceful labelings in the context of graph games. The Graceful game was introduced by Tuza in 2017 as a two-players game on a connected graph in which the players Alice and Bob take turns labeling the vertices with distinct integers from 0 to $m$. Alice's goal is to gracefully label the graph as Bob's goal is to prevent it from happening. In this work, we study winning strategies for Alice and Bob in complete graphs, paths, cycles, complete bipartite graphs, caterpillars, prisms, wheels, helms, webs, gear graphs, hypercubes and some powers of paths

Jogo “você é o(a) técnico(a)”: praticando conceitos de permutação, arranjo e combinação

Neste trabalho, apresentamos as potencialidades da proposta didática de jogos no ensino de matemática, em particular de análise combinatória. Propomos o jogo “Você é o(a) Técnico”, que possui como principal objetivo praticar os conceitos de permutação, arranjo e combinação, assim como estabelecer a relação e a diferença que tais conceitos possuem entre si. Apresentamos os resultados de uma pesquisa em relação à análise combinatória como disciplina obrigatória em treze cursos de formação de professor do Rio de Janeiro. Com base na pesquisa, entrevistamos o coordenador do curso de Licenciatura em Matemática da Universidade Federal Fluminense a respeito da adoção da análise combinatória na grade obrigatória do curso após a mudança de currículo, ocorrida em 2018; e, sobre o ensino de análise combinatória nas escolas. Apresentamos referenciais teóricos que tentam explicar porque a combinatória ainda é ensinada como um mero jogo de fórmulas, e mostrar novas formas de ensinar a disciplina valorizando o raciocínio combinatórioIn this work, we present the potentialities of using games in math teaching, particularly combinatory. We propose the game “Você é o(a) Técnico(a)”, whose main objective is to practice knowledge about permutation, arrangement and combination, such as establish the difference between such concepts. We present the results from a research regarding the combinatorial analysis as a compulsory subject in thirteen teacher-training courses in Rio de Janeiro. Based on this research, we interviewed the coordinator of the Mathematics Degree course from Fluminense Federal University about the adoption of combinatory among the compulsory subjects after the curriculum change, in 2018, and, about the combinatorial teaching in schools. We present theoretical references that tries to explain why the combinatorial analysis is still taught as mathematical formulas, and show new ways of teaching this subject valuing the combinatorial reasoning36 p

On the Graceful Game

A graceful labeling of a graph G with m edges consists in labeling the vertices of G with distinct integers from 0 to m such that, when each edge is assigned the absolute difference of the labels of its endpoints, all induced edge labels are distinct. Rosa established two well known conjectures: all trees are graceful (1966) and all triangular cacti are graceful (1988). In order to contribute to both conjectures we study these problems in the context of graph games. The graceful game was introduced by Tuza in 2017 as a two-players game on a connected graph in which the players Alice and Bob take turns labeling the vertices with distinct integers from 0 to m. Alice’s goal is to gracefully label the graph as Bob’s goal is to prevent it from happening. In this work, we present the first results in this area by showing winning strategies for Alice and Bob in complete graphs, paths, cycles, complete bipartite graphs, caterpillars, prisms, wheels, helms, webs, gear graphs, hypercubes and some powers of paths

On the Graceful Game

A graceful labeling of a graph G with m edges consists in labeling the vertices of G with distinct integers from 0 to m such that, when each edge is assigned the absolute difference of the labels of its endpoints, all induced edge labels are distinct. Rosa established two well known conjectures: all trees are graceful (1966) and all triangular cacti are graceful (1988). In order to contribute to both conjectures we study these problems in the context of graph games. The graceful game was introduced by Tuza in 2017 as a two-players game on a connected graph in which the players Alice and Bob take turns labeling the vertices with distinct integers from 0 to m. Alice’s goal is to gracefully label the graph as Bob’s goal is to prevent it from happening. In this work, we present the first results in this area by showing winning strategies for Alice and Bob in complete graphs, paths, cycles, complete bipartite graphs, caterpillars, prisms, wheels, helms, webs, gear graphs, hypercubes and some powers of paths

Graceful game on some graph classes

A graceful labeling of a graph $G$ with $m$ edges consists in labeling the vertices of $G$ with distinct integers from $0$ to $m$ such that each edge is uniquely identified by the absolute difference of the labels of its endpoints. In this work, we study the graceful labeling problem in the context of maker-breaker graph games. The Graceful Game was introduced by Tuza, in 2017, as a two-players game on a connected graph in which the players, Alice and Bob, take moves labeling the vertices with distinct integers from $0$ to $m$. Players are constrained to use only legal labelings (moves), that is, after a move, all edge labels are distinct. Alice's goal is to obtain a graceful labeling for the graph, as Bob's goal is to prevent it from happening. In this work, we study winning strategies for Alice and Bob in graph classes: paths, complete graphs, cycles, complete bipartite graphs, caterpillars, trees, gear graphs, web graphs, prisms, hypercubes,  2-powers of paths, wheels and fan graphs