Results on the graceful game and range-relaxed graceful Game on graphs and new variants of labeling games

Detalhes bibliográficos
Autor(a) principal: Oliveira, Deise Lilian de
Data de Publicação: 2024
Tipo de documento: Tese
Idioma: eng
Título da fonte: Repositório Institucional da Universidade Federal Fluminense (RIUFF)
Texto Completo: https://app.uff.br/riuff/handle/1/33037
Resumo: Given a graph G = (V, E) and a set of consecutive integer labels L = {0, . . . , k}, a vertex labeling of G is a function f : V (G) −→ L that induces an edge labeling g(uv) for every edge uv ∈ E(G). When the function f is injective and the label induced on each edge uv ∈ E(G) is given by g(uv) = |f(u) − f(v)|, resulting in all edge labels being distinct, we call the labeling graceful, if additionally, k = |E(G)| or Range-Relaxed Graceful when this condition is relaxed, that is, k ≥ |E(G)|. Since 1963, many graph labelings based on sums of integer labels have been proposed. Thus, if the vertex labeling f is injective and the label induced on the edges is given by g(uv) = f(u) + f(v), then the labeling is called Edge-Sum Distinguishing (ESD). Another way to approach the topic of graph labeling is through combinatorial games. In 2017, Tuza published an article emphasizing that despite the graph labeling area being a vast area of work, with several open problems, little had been done in combinatorial games so far. Tuza proposed the study of new maker-breaker games such as the graceful game, the Range-Relaxed Graceful game (RRG game) and the Edge-Sum Distinguishing game (ESD game). In these games, Alice and Bob alternately assign an unused label (respecting the rules of the labeling that is being constructed) to a vertex not labeled. Alice’s goal is to finish the game with a vertex labeling of G and Bob’s goal is to prevent this from happening. Tuza also posed the following questions about the labeling games: given a simple graph G and an integer k, indicating the maximum value of the set of consecutive integer labels, for which values of k can Alice win the game? And if Alice wins the game with the set of labels up to k, can she also win with labels up to k + 1? In this work, we partially answer these questions for the graceful game, RRG game and ESD game, by presenting limits to the number of consecutive nonnegative integer labels L = {0, . . . , k} needed for Alice to win the game on classic families of graph. We investigate the graceful game on Cartesian products and corona products. In addition, we present the first results in the literature on the RRG game, and we proved that Alice wins on any graph G with order n for any set L = {0, 1, . . . , k} with k ≥ (n − 1) + ∆(∆−1) 2 + max{d(v) · 2(m − d(v)): v ∈ V (G)}, which gives affirmative answers for the two questions posed by Tuza. Futhermore, we present bounds for the ESD game number, that is the minimum nonnegative integer k such that Alice has a winning strategy playing the ESD game on G with a set of labels L = {0, . . . , k}, independently of which player starts the gam
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spelling Results on the graceful game and range-relaxed graceful Game on graphs and new variants of labeling gamesgraph labelinglabeling gamesJogo em Educação MatemáticaGraforotulação de grafosjogos de rotulaçãoGiven a graph G = (V, E) and a set of consecutive integer labels L = {0, . . . , k}, a vertex labeling of G is a function f : V (G) −→ L that induces an edge labeling g(uv) for every edge uv ∈ E(G). When the function f is injective and the label induced on each edge uv ∈ E(G) is given by g(uv) = |f(u) − f(v)|, resulting in all edge labels being distinct, we call the labeling graceful, if additionally, k = |E(G)| or Range-Relaxed Graceful when this condition is relaxed, that is, k ≥ |E(G)|. Since 1963, many graph labelings based on sums of integer labels have been proposed. Thus, if the vertex labeling f is injective and the label induced on the edges is given by g(uv) = f(u) + f(v), then the labeling is called Edge-Sum Distinguishing (ESD). Another way to approach the topic of graph labeling is through combinatorial games. In 2017, Tuza published an article emphasizing that despite the graph labeling area being a vast area of work, with several open problems, little had been done in combinatorial games so far. Tuza proposed the study of new maker-breaker games such as the graceful game, the Range-Relaxed Graceful game (RRG game) and the Edge-Sum Distinguishing game (ESD game). In these games, Alice and Bob alternately assign an unused label (respecting the rules of the labeling that is being constructed) to a vertex not labeled. Alice’s goal is to finish the game with a vertex labeling of G and Bob’s goal is to prevent this from happening. Tuza also posed the following questions about the labeling games: given a simple graph G and an integer k, indicating the maximum value of the set of consecutive integer labels, for which values of k can Alice win the game? And if Alice wins the game with the set of labels up to k, can she also win with labels up to k + 1? In this work, we partially answer these questions for the graceful game, RRG game and ESD game, by presenting limits to the number of consecutive nonnegative integer labels L = {0, . . . , k} needed for Alice to win the game on classic families of graph. We investigate the graceful game on Cartesian products and corona products. In addition, we present the first results in the literature on the RRG game, and we proved that Alice wins on any graph G with order n for any set L = {0, 1, . . . , k} with k ≥ (n − 1) + ∆(∆−1) 2 + max{d(v) · 2(m − d(v)): v ∈ V (G)}, which gives affirmative answers for the two questions posed by Tuza. Futhermore, we present bounds for the ESD game number, that is the minimum nonnegative integer k such that Alice has a winning strategy playing the ESD game on G with a set of labels L = {0, . . . , k}, independently of which player starts the gamDado um grafo G = (V,E) e um conjunto de rótulos inteiros consecutivos L = {0, . . . , k}, uma rotulação de vértices de G é uma função f : V (G) −→ L que induz uma rotulação de arestas g(uv) para toda aresta uv ∈ E(G). Quando a função f é injetiva e o rótulo induzido em cada aresta uv ∈ E(G) é dado por g(uv) = |f (u) − f (v)|, resultando em todos os rótulos de arestas distintos, chamamos a rotulação f de graceful, se adicionalmente, k = |E(G)|, ou Range-Relaxed Graceful (RRG) quando essa condição é relaxada, ou seja, k ≥ |E(G)|. Desde 1963, muitas rotulações de grafos baseadas em somas de rótulos inteiros foram propostas. Assim, se a rotulação de vértices f é injetiva e o rótulo induzido nas arestas é dado por g(uv) = f (u) + f (v), então a rotulação é chamada de Edge-Sum Distinguishing (ESD). Uma outra maneira de abordar o tema de rotulação é através de jogos combinatórios. Em 2017, Tuza publicou um artigo enfatizando que apesar da rotulação de grafos ser uma área vasta de trabalho, com diversos problemas em aberto, pouco havia sido feito em jogos combinatórios até o momento. Tuza propôs o estudo de novos jogos maker-breaker tais como o jogo graceful, o jogo Range-Relaxed Graceful (RRG) e o jogo Edge-Sum Distinguishing (ESD). Nesses jogos, Alice e Bob atribuem alternadamente um rótulo (respeitando as regras da rotulação que está sendo construída) não usado a um vértice não rotulado. O objetivo da Alice é terminar o jogo com uma rotulação de vértices de G e o objetivo do Bob é impedir que isso aconteça. Tuza também formulou as seguintes questões sobre os jogos de rotulação: dado um grafo simples G e um inteiro k, que indica o valor máximo do conjunto de rótulos inteiros consecutivos, para quais valores de k Alice pode vencer o jogo? E se Alice vencer o jogo com o conjunto de rótulos até k, ela também poderá ganhar com rótulos até k + 1? Nesse trabalho, respondemos a essas questões para os jogos graceful, RRG e ESD, apresentando limites para o número de rótulos inteiros consecutivos não negativos L = {0, . . . , k} necessários para Alice vencer o jogo em famílias clássicas de grafos. Investigamos o jogo graceful em produtos Cartesiano e produto corona de grafos. Adicionalmente, apresentamos os primeiros resultados na literatura sobre o jogo RRG, e provamos que Alice vence em qualquer grafo G com ordem n para ∆(∆−1) qualquer conjunto L = {0, 1, . . . , k} com k ≥ (n − 1) + max{d(v) · 2(m − d(v)) : v ∈ 2 V (G)}, respondendo de modo afirmativo às duas questões colocadas por Tuza. Além disso, apresentamos limites para o número do jogo ESD, que é o mínimo número inteiro não negativo k tal que Alice tem uma estratégia vencedora jogando em um grafo G com um conjunto de rótulos L = {0, . . . , k}, independentemente de qual jogador inicia o jogo.81 f.Souza, Simone Dantas deLuiz, Atílio GomesOliveira, Deise Lilian de2024-07-05T19:34:20Z2024-07-05T19:34:20Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisapplication/pdfOLIVEIRA, Deise Lilian de. Results on the Graceful Game and Range-Relaxed Graceful Game on Graphs and new variants of labeling games. 2023. 81 f. Tese (Doutorado em Matemática) – Programa de Pós-Graduação em Matemática, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, 2023.https://app.uff.br/riuff/handle/1/33037CC-BY-SAinfo:eu-repo/semantics/openAccessengreponame:Repositório Institucional da Universidade Federal Fluminense (RIUFF)instname:Universidade Federal Fluminense (UFF)instacron:UFF2024-07-05T19:34:24Zoai:app.uff.br:1/33037Repositório InstitucionalPUBhttps://app.uff.br/oai/requestriuff@id.uff.bropendoar:21202024-08-19T11:06:40.181481Repositório Institucional da Universidade Federal Fluminense (RIUFF) - Universidade Federal Fluminense (UFF)false
dc.title.none.fl_str_mv Results on the graceful game and range-relaxed graceful Game on graphs and new variants of labeling games
title Results on the graceful game and range-relaxed graceful Game on graphs and new variants of labeling games
spellingShingle Results on the graceful game and range-relaxed graceful Game on graphs and new variants of labeling games
Oliveira, Deise Lilian de
graph labeling
labeling games
Jogo em Educação Matemática
Grafo
rotulação de grafos
jogos de rotulação
title_short Results on the graceful game and range-relaxed graceful Game on graphs and new variants of labeling games
title_full Results on the graceful game and range-relaxed graceful Game on graphs and new variants of labeling games
title_fullStr Results on the graceful game and range-relaxed graceful Game on graphs and new variants of labeling games
title_full_unstemmed Results on the graceful game and range-relaxed graceful Game on graphs and new variants of labeling games
title_sort Results on the graceful game and range-relaxed graceful Game on graphs and new variants of labeling games
author Oliveira, Deise Lilian de
author_facet Oliveira, Deise Lilian de
author_role author
dc.contributor.none.fl_str_mv Souza, Simone Dantas de
Luiz, Atílio Gomes
dc.contributor.author.fl_str_mv Oliveira, Deise Lilian de
dc.subject.por.fl_str_mv graph labeling
labeling games
Jogo em Educação Matemática
Grafo
rotulação de grafos
jogos de rotulação
topic graph labeling
labeling games
Jogo em Educação Matemática
Grafo
rotulação de grafos
jogos de rotulação
description Given a graph G = (V, E) and a set of consecutive integer labels L = {0, . . . , k}, a vertex labeling of G is a function f : V (G) −→ L that induces an edge labeling g(uv) for every edge uv ∈ E(G). When the function f is injective and the label induced on each edge uv ∈ E(G) is given by g(uv) = |f(u) − f(v)|, resulting in all edge labels being distinct, we call the labeling graceful, if additionally, k = |E(G)| or Range-Relaxed Graceful when this condition is relaxed, that is, k ≥ |E(G)|. Since 1963, many graph labelings based on sums of integer labels have been proposed. Thus, if the vertex labeling f is injective and the label induced on the edges is given by g(uv) = f(u) + f(v), then the labeling is called Edge-Sum Distinguishing (ESD). Another way to approach the topic of graph labeling is through combinatorial games. In 2017, Tuza published an article emphasizing that despite the graph labeling area being a vast area of work, with several open problems, little had been done in combinatorial games so far. Tuza proposed the study of new maker-breaker games such as the graceful game, the Range-Relaxed Graceful game (RRG game) and the Edge-Sum Distinguishing game (ESD game). In these games, Alice and Bob alternately assign an unused label (respecting the rules of the labeling that is being constructed) to a vertex not labeled. Alice’s goal is to finish the game with a vertex labeling of G and Bob’s goal is to prevent this from happening. Tuza also posed the following questions about the labeling games: given a simple graph G and an integer k, indicating the maximum value of the set of consecutive integer labels, for which values of k can Alice win the game? And if Alice wins the game with the set of labels up to k, can she also win with labels up to k + 1? In this work, we partially answer these questions for the graceful game, RRG game and ESD game, by presenting limits to the number of consecutive nonnegative integer labels L = {0, . . . , k} needed for Alice to win the game on classic families of graph. We investigate the graceful game on Cartesian products and corona products. In addition, we present the first results in the literature on the RRG game, and we proved that Alice wins on any graph G with order n for any set L = {0, 1, . . . , k} with k ≥ (n − 1) + ∆(∆−1) 2 + max{d(v) · 2(m − d(v)): v ∈ V (G)}, which gives affirmative answers for the two questions posed by Tuza. Futhermore, we present bounds for the ESD game number, that is the minimum nonnegative integer k such that Alice has a winning strategy playing the ESD game on G with a set of labels L = {0, . . . , k}, independently of which player starts the gam
publishDate 2024
dc.date.none.fl_str_mv 2024-07-05T19:34:20Z
2024-07-05T19:34:20Z
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/doctoralThesis
format doctoralThesis
status_str publishedVersion
dc.identifier.uri.fl_str_mv OLIVEIRA, Deise Lilian de. Results on the Graceful Game and Range-Relaxed Graceful Game on Graphs and new variants of labeling games. 2023. 81 f. Tese (Doutorado em Matemática) – Programa de Pós-Graduação em Matemática, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, 2023.
https://app.uff.br/riuff/handle/1/33037
identifier_str_mv OLIVEIRA, Deise Lilian de. Results on the Graceful Game and Range-Relaxed Graceful Game on Graphs and new variants of labeling games. 2023. 81 f. Tese (Doutorado em Matemática) – Programa de Pós-Graduação em Matemática, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, 2023.
url https://app.uff.br/riuff/handle/1/33037
dc.language.iso.fl_str_mv eng
language eng
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dc.source.none.fl_str_mv reponame:Repositório Institucional da Universidade Federal Fluminense (RIUFF)
instname:Universidade Federal Fluminense (UFF)
instacron:UFF
instname_str Universidade Federal Fluminense (UFF)
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reponame_str Repositório Institucional da Universidade Federal Fluminense (RIUFF)
collection Repositório Institucional da Universidade Federal Fluminense (RIUFF)
repository.name.fl_str_mv Repositório Institucional da Universidade Federal Fluminense (RIUFF) - Universidade Federal Fluminense (UFF)
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