We study the (a,b)-monochromatic transversal game that is a combinatorial Maker-Breaker game where Alice and Bob alternately colour a vertices in red and b vertices in blue of a hypergraph, respectively. Either player is enabled to start the game. Alice tries to construct a hyperedge transversal, and Bob tries to prevent this. The winner is Alice if she obtains a red hyperedge transversal; otherwise, Bob wins the game if he obtains a monochromatic blue hyperedge. Maker-breaker games were determined to be PSPACE-complete. In this work, we analyze the game played on clique-hypergraphs of powers of cycles, and we show strategies that, depending on the choice of the parameters, allow a specific player to win the game.
{"title":"The (a,b)-monochromatic transversal game on clique-hypergraphs of powers of cycles","authors":"Wilder Mendes, Simone Dantas, Sylvain Gravier","doi":"10.1051/ro/2024054","DOIUrl":"https://doi.org/10.1051/ro/2024054","url":null,"abstract":"We study the (a,b)-monochromatic transversal game that is a combinatorial Maker-Breaker game where Alice and Bob alternately colour a vertices in red and b vertices in blue of a hypergraph, respectively. Either player is enabled to start the game. Alice tries to construct a hyperedge transversal, and Bob tries to prevent this. The winner is Alice if she obtains a red hyperedge transversal; otherwise, Bob wins the game if he obtains a monochromatic blue hyperedge. Maker-breaker games were determined to be PSPACE-complete. In this work, we analyze the game played on clique-hypergraphs of powers of cycles, and we show strategies that, depending on the choice of the parameters, allow a specific player to win the game.","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"37 20","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140262924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In order to solve the capital-constrained enterprises to promote green R&D and industrial green transformation, taking into account risk aversion behavior and capital shortage factors, the paper considers a green supply chain composed of one retailer, one common manufacturer, and one capital-constrained green manufacturer and develops the different models of risk aversion and competition to explore the optimal financing strategies under the different financing modes. The results show that: (1) When the financing interest rates are equal, the green manufacturer should prefer the retailer financing mode, and regardless of the financing method, the increase of financing interest rate is extremely unfavorable to the retailer. (2) The increase in the degree of green manufacturer’s risk aversion is not conducive to the long-term development of the common manufacturer and himself, and it is beneficial to the retailer, but it will cause market fluctuations, while the increase in the degree of the retailer’s risk aversion is only detrimental to himself, but beneficial for supply chain members.
{"title":"Green financing strategies under risk aversion and manufacturer competition","authors":"Guangdong Liu, Jinggui Chen, Ziyang Li","doi":"10.1051/ro/2024052","DOIUrl":"https://doi.org/10.1051/ro/2024052","url":null,"abstract":"In order to solve the capital-constrained enterprises to promote green R&D and industrial green transformation, taking into account risk aversion behavior and capital shortage factors, the paper considers a green supply chain composed of one retailer, one common manufacturer, and one capital-constrained green manufacturer and develops the different models of risk aversion and competition to explore the optimal financing strategies under the different financing modes. The results show that: (1) When the financing interest rates are equal, the green manufacturer should prefer the retailer financing mode, and regardless of the financing method, the increase of financing interest rate is extremely unfavorable to the retailer. (2) The increase in the degree of green manufacturer’s risk aversion is not conducive to the long-term development of the common manufacturer and himself, and it is beneficial to the retailer, but it will cause market fluctuations, while the increase in the degree of the retailer’s risk aversion is only detrimental to himself, but beneficial for supply chain members.","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"5 7","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140410518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Modern requirements necessitate the establishment of sustainable transportation systems, considering the substantial growth in transportation activities over recent years, which is expected to continue. Companies are facing the challenge of modeling their system transport to align with green principles. Sustainable transport relied on involving diverse stakeholders, particularly scientific research, in the development of this field. In light of this, maintaining sustainable transport quality involves conducting thorough investigations into an innovative study focusing on an uncertain interval programming model for a multi-stage, multiobjective, multi-product transportation challenge within budget constraints and safety measures in a green supply chain. Human languages often contain imperfect or unknown information, inherently lacking certainty; achieving precision in describing existing states or future outcomes is frequently unattainable. In probability theory, sufficient historical information is crucial for estimating probability distributions; while in fuzzy theory, determining a reliable membership function proves challenging; hence, there is often a hesitant estimation of the degree of belief in the occurrence of each condition. Addressing such uncertainties, the theory of uncertain intervals proves highly valuable. Given these considerations, the elements of the specified problem are recognized as uncertain intervals. To manage this lack of assurance, a fusion of interval theory and methods from uncertain programming is used to formulate two distinct models: an expected value model and a chance-constrained model. The equivalent deterministic models are then formulated and solved utilizing Weighted Sum Method, fuzzy programming, and goal programming. Following this, a numerical example is utilized to assess the model’s performance, and the results obtained are compared. Finally, the document concludes with a sensitivity analysis and outlines future directions
{"title":"Beyond green borders: an innovative model for\u0000sustainable transportation in supply chains","authors":"Sifaoui Thiziri, Méziane Aider","doi":"10.1051/ro/2024053","DOIUrl":"https://doi.org/10.1051/ro/2024053","url":null,"abstract":"Modern requirements necessitate the establishment of sustainable transportation systems, considering the substantial growth in transportation activities over\u0000recent years, which is expected to continue. Companies are facing the challenge\u0000of modeling their system transport to align with green principles. Sustainable\u0000transport relied on involving diverse stakeholders, particularly scientific research,\u0000in the development of this field. In light of this, maintaining sustainable transport quality involves conducting thorough investigations into an innovative study\u0000focusing on an uncertain interval programming model for a multi-stage, multiobjective, multi-product transportation challenge within budget constraints and\u0000safety measures in a green supply chain. Human languages often contain imperfect or unknown information, inherently lacking certainty; achieving precision in\u0000describing existing states or future outcomes is frequently unattainable. In probability theory, sufficient historical information is crucial for estimating probability\u0000distributions; while in fuzzy theory, determining a reliable membership function\u0000proves challenging; hence, there is often a hesitant estimation of the degree of\u0000belief in the occurrence of each condition. Addressing such uncertainties, the theory of uncertain intervals proves highly valuable. Given these considerations, the\u0000elements of the specified problem are recognized as uncertain intervals. To manage this lack of assurance, a fusion of interval theory and methods from uncertain\u0000programming is used to formulate two distinct models: an expected value model\u0000and a chance-constrained model. The equivalent deterministic models are then formulated and solved utilizing Weighted Sum Method, fuzzy programming, and goal programming. Following this, a numerical example is utilized to assess the model’s performance, and the results obtained are compared. Finally, the document concludes with a sensitivity analysis and outlines future directions","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"19 8","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140411659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Maojun Zhang, Lu Shen, Jiangxia Nan, Jizhou Wang, Zhonghang Xia, Yang Zhao
In this paper, we explore optimal strategies of a supply chain consisting of one manufacturer, one capital-constrained retailer and one bank, where the bank provides loans to the retailer due to credit guarantees. However, there are default risks if the retailer can not repay the loans. Using conditional value-at-risk (CVaR) to describe risk aversion of the retailer, optimal order quantities of the retailer and optimal wholesale prices of the manufacturer are obtained by solving a Stackelberg game model, where the manufacturer is a leader and the retailer is a follower, respectively. Our numerical results show that default probabilities of the retailer are proportional to optimal wholesale prices of the manufacturer. It implies that when the default probability of the retailer is high, the manufacturer should reduce the default risk by setting higher wholesale prices to avoid a burden of a substantial guarantee. Thus, our results can serve as insights for decision-makers of supply chain management.
{"title":"Optimal strategy for supply chain with credit guarantee using CVaR","authors":"Maojun Zhang, Lu Shen, Jiangxia Nan, Jizhou Wang, Zhonghang Xia, Yang Zhao","doi":"10.1051/ro/2024051","DOIUrl":"https://doi.org/10.1051/ro/2024051","url":null,"abstract":"In this paper, we explore optimal strategies of a supply chain consisting of one manufacturer, one capital-constrained retailer and one bank, where the bank provides loans to the retailer due to credit guarantees. However, there are default risks if the retailer can not repay the loans. Using conditional value-at-risk (CVaR) to describe risk aversion of the retailer, optimal order quantities of the retailer and optimal wholesale prices of the manufacturer are obtained by solving a Stackelberg game model, where the manufacturer is a leader and the retailer is a follower, respectively. Our numerical results show that default probabilities of the retailer are proportional to optimal wholesale prices of the manufacturer. It implies that when the default probability of the retailer is high, the manufacturer should reduce the default risk by setting higher wholesale prices to avoid a burden of a substantial guarantee. Thus, our results can serve as insights for decision-makers of supply chain management.","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"25 11","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140424459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
With the continuous improvement of product environmental standards, using or selling older generation products will increase additional environmental costs, resulting in a decrease in consumer preference for older generation products or products on hand. This paper investigates the impact of specific product environmental standards implementation on enterprise product line extension and pricing strategies. We find that if the production cost is low or the consumers' green sensitivity is high enough, the manufacturer's green production can be better than the designated standard. When the unit production cost of new products is within a certain range, the manufacturer's profit will increase, otherwise it will decrease. In addition, we present the manufacturer's product line update strategy in different market segments defined by different cost thresholds, which indicate the cases where the manufacturer will be forced to withdraw from the market. Moreover, we examine the correlation between consumer quality preference and market demand, and discover that an elevation in consumer preference for product functional quality does not necessarily result in a corresponding increase in product demand. Finally, we investigate the relationship between the manufacturer's actual green product decision and the specified environmental standard, and give the decision areas where the manufacturer's actual green decision is higher (or lower) than or equal to the specified green standard. The results suggest that blindly improving environmental standards by policymakers does not necessarily lead to an improvement in manufacturers' green decisions.
{"title":"Green innovation and product line decisions under environmental standard upgrading","authors":"Miaomiao Wang, Xinyu Chen, Xiaoxi Zhu, Kai Liu","doi":"10.1051/ro/2024050","DOIUrl":"https://doi.org/10.1051/ro/2024050","url":null,"abstract":"With the continuous improvement of product environmental standards, using or selling older generation products will increase additional environmental costs, resulting in a decrease in consumer preference for older generation products or products on hand. This paper investigates the impact of specific product environmental standards implementation on enterprise product line extension and pricing strategies. We find that if the production cost is low or the consumers' green sensitivity is high enough, the manufacturer's green production can be better than the designated standard. When the unit production cost of new products is within a certain range, the manufacturer's profit will increase, otherwise it will decrease. In addition, we present the manufacturer's product line update strategy in different market segments defined by different cost thresholds, which indicate the cases where the manufacturer will be forced to withdraw from the market. Moreover, we examine the correlation between consumer quality preference and market demand, and discover that an elevation in consumer preference for product functional quality does not necessarily result in a corresponding increase in product demand. Finally, we investigate the relationship between the manufacturer's actual green product decision and the specified environmental standard, and give the decision areas where the manufacturer's actual green decision is higher (or lower) than or equal to the specified green standard. The results suggest that blindly improving environmental standards by policymakers does not necessarily lead to an improvement in manufacturers' green decisions.","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"39 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140435865","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Usha Aggarwal, Mukesh Kumar Mehlawat, Pankaj Gupta, Vincent Charles
This paper aims to establish a supply chain model that significantly reduces economic and environmental costs. It comprises all activities related to procurement, production, and distribution planning. The proposed multi-objective multi-commodity optimisation model deals with the four conflicting objectives of reducing costs and emissions and choosing top-priority suppliers and the most efficient vehicles. We apply an integrated AHP (analytic hierarchy process) and TOPSIS (technique for order preference by similarity to an ideal solution) technique to determine the weights of suppliers, depending on three indices of criteria, alternatives, and raw material. This paper proposes a cross-efficiency evaluation method using data envelopment analysis (DEA) to ensure that the cross-evaluation of different types of vehicles for evaluating peers is as consistent as possible. The mutually contradictory objectives give rise to several Pareto-optimal solutions. The optimal compromise solutions are found using a lexicographic goal programming technique. We present a real-world case related to the paper industry to demonstrate the effectiveness of the proposed methodology, followed by numerical comparisons and additional insights.
{"title":"An optimisation model for sustainable multi-commodity transportation planning","authors":"Usha Aggarwal, Mukesh Kumar Mehlawat, Pankaj Gupta, Vincent Charles","doi":"10.1051/ro/2024049","DOIUrl":"https://doi.org/10.1051/ro/2024049","url":null,"abstract":"This paper aims to establish a supply chain model that significantly reduces economic and environmental costs. It comprises all activities related to procurement, production, and distribution planning. The proposed multi-objective multi-commodity optimisation model deals with the four conflicting objectives of reducing costs and emissions and choosing top-priority suppliers and the most efficient vehicles. We apply an integrated AHP (analytic hierarchy process) and TOPSIS (technique for order preference by similarity to an ideal solution) technique to determine the weights of suppliers, depending on three indices of criteria, alternatives, and raw material. This paper proposes a cross-efficiency evaluation method using data envelopment analysis (DEA) to ensure that the cross-evaluation of different types of vehicles for evaluating peers is as consistent as possible. The mutually contradictory objectives give rise to several Pareto-optimal solutions. The optimal compromise solutions are found using a lexicographic goal programming technique. We present a real-world case related to the paper industry to demonstrate the effectiveness of the proposed methodology, followed by numerical comparisons and additional insights.","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"23 19","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140442072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The maximum distance matrix of a strongly connected digraph is a symmetric matrix whose rows and columns are indexed the vertices, the entries of which correspond to the maximum directed distance between the vertices. In this paper, we determine the digraphs that uniquely minimize the largest eigenvalue of the maximum distance matrix in some classes of strongly connected digraphs, and the $n$-vertex strongly connected digraphs for which the maximum distance matrices have an eigenvalue with multiplicity $n-1$.
{"title":"On spectral properties of digraphs about maximum distance","authors":"Le Xu, Shilin Wang, Bo Zhou","doi":"10.1051/ro/2024048","DOIUrl":"https://doi.org/10.1051/ro/2024048","url":null,"abstract":"The maximum distance matrix of a strongly connected digraph is a symmetric matrix whose rows and columns are indexed the vertices, the entries of which correspond to the maximum directed distance between the vertices. In this paper, we determine the digraphs that uniquely minimize the largest eigenvalue of the maximum distance matrix in some classes of strongly connected digraphs, and the $n$-vertex strongly connected digraphs for which the maximum distance matrices have an eigenvalue with multiplicity $n-1$.","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"348 ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140448065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper estimates the Most Productive Scale Size (MPSS) in the NDEA model to appraise the sustainability and resilience of the supply chains. As the corresponding input and output criteria are not always accurately measurable, we also introduce the fuzzy version of our proposed NDEA model and apply the proposed model in a case study involving 10 Iranian supply chains of Companies Producing Soft Drinks (CPSDs). The considered-three-echelon supply chains include suppliers, manufacturers, and distributors. Mathematical analysis proves that the MPSS of the considered supply chain can be decomposed as the sum of the MPSS values of the individual stages. Thus, the supply chain is overall MPSS if and only if it is MPSS in every three stages. The results of this study reveal that the Behnoush supply chain is overall MPSS in all three stages, including supplier, manufacturer, and distributor, for any α ∈ {0.1,0.5, 1}. A sensitivity analysis has been performed to measure the impact of each criterion on the entire supply chain performance. The sensitivity analysis results indicate that the social and resilience criteria significantly impact the performance and ranking of supply chains. Finally, we discuss how to improve the sustainability and resilience of non-MPSS supply chains.
{"title":"Estimating most productive scale size decomposition in a fuzzy network data envelopment analysis model: assessing the sustainability and resilience of the supply chain","authors":"Mohammad Tavassoli, Mahsa Ghandehari","doi":"10.1051/ro/2024047","DOIUrl":"https://doi.org/10.1051/ro/2024047","url":null,"abstract":"This paper estimates the Most Productive Scale Size (MPSS) in the NDEA model to appraise the sustainability and resilience of the supply chains. As the corresponding input and output criteria are not always accurately measurable, we also introduce the fuzzy version of our proposed NDEA model and apply the proposed model in a case study involving 10 Iranian supply chains of Companies Producing Soft Drinks (CPSDs). The considered-three-echelon supply chains include suppliers, manufacturers, and distributors. Mathematical analysis proves that the MPSS of the considered supply chain can be decomposed as the sum of the MPSS values of the individual stages. Thus, the supply chain is overall MPSS if and only if it is MPSS in every three stages. The results of this study reveal that the Behnoush supply chain is overall MPSS in all three stages, including supplier, manufacturer, and distributor, for any α ∈ {0.1,0.5, 1}. A sensitivity analysis has been performed to measure the impact of each criterion on the entire supply chain performance. The sensitivity analysis results indicate that the social and resilience criteria significantly impact the performance and ranking of supply chains. Finally, we discuss how to improve the sustainability and resilience of non-MPSS supply chains.","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"536 ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140446908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Diane Castonguay, Celina de Figueiredo, Luis Kowada, C. Patrão, Diana Sasaki, Mario Valencia-Pabon
A textit{$k$-total coloring} of a graph $G$ is an assignment of $k$ colors to the elements (vertices and edges) of $G$ so that adjacent or incident elements have different colors. The total chromatic number is the smallest integer $k$ for which $G$ has a $k$-total coloring. The well known Total Coloring Conjecture states that the total chromatic number of a graph is either $Delta(G)+1$ (called Type~1) or $Delta(G)+2$ (called Type~2), where $Delta(G)$ is the maximum degree of $G$. We consider the direct product of complete graphs $K_m times K_n$. It is known that if at least one of the numbers $m$ or $n$ is even, then $K_m times K_n$ is Type~1, except for $K_2 timesK_2$. We prove that the graph $K_m times K_n$ is Type~1 when both $m$ and $n$ are odd numbers, by using that the conformable condition is sufficient for the graph $K_m times K_n$ to be Type~1 when both $m$ and $n$ are large enough, and by constructing the target total colorings by using Hamiltonian decompositions and a specific color class, called guiding color. We additionally apply our technique to the direct product $C_m times K_n$ of a cycle with a complete graph. Interestingly, we are able to find a Type 2 infinite family $C_m times K_n$, when $m$ is not a multiple of 3 and $n = 2$. We provide evidence to conjecture that all other $C_m times K_n$ are Type 1.
{"title":"On the total chromatic number of the direct product of cycles and complete graphs","authors":"Diane Castonguay, Celina de Figueiredo, Luis Kowada, C. Patrão, Diana Sasaki, Mario Valencia-Pabon","doi":"10.1051/ro/2024045","DOIUrl":"https://doi.org/10.1051/ro/2024045","url":null,"abstract":"A textit{$k$-total coloring} of a graph $G$ is an assignment of $k$ colors to the elements (vertices and edges) of $G$ so that adjacent or incident elements have different colors. The total chromatic number is the smallest integer $k$ for which $G$ has a $k$-total coloring. The well known Total Coloring Conjecture states that the total chromatic number of a graph is either $Delta(G)+1$ (called Type~1) or $Delta(G)+2$ (called Type~2), where $Delta(G)$ is the maximum degree of $G$. \u0000We consider the direct product of complete graphs $K_m times K_n$. \u0000It is known that if at least one of the numbers $m$ or $n$ is even, then $K_m times K_n$ is Type~1, except for $K_2 timesK_2$. \u0000We prove that the graph $K_m times K_n$ is Type~1 when both $m$ and $n$ are odd numbers, by using that the conformable condition is sufficient for the graph $K_m times K_n$ to be Type~1 when both $m$ and $n$ are large enough, and by constructing the target total colorings by using Hamiltonian decompositions and a specific color class, called guiding color. \u0000We additionally apply our technique to the direct product $C_m times K_n$ of a cycle with a complete graph.\u0000Interestingly, we are able to find a Type 2 infinite family $C_m times K_n$, when $m$ is not a multiple of 3 and $n = 2$.\u0000We provide evidence to conjecture that all other $C_m times K_n$ are Type 1.","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"99 16","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140451459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rudini Sampaio, Gabriel A. G. Sobral, Yoshiko Wakabayashi
In a graph $G$, a set $Csubseteq V(G)$ is an identifying code if, for all vertices $v$ in $G$, the sets $N[v]cap C$ are all nonempty and pairwise distinct, where $N[v]$ denotes the closed neighbourhood of $v$. We focus on the minimum density of identifying codes of infinite hexagonal grids $H_k$ with $k$ rows, denoted by $d^*(H_k)$, and present optimal solutions for $kleq 5$. Using discharging method, we also prove a lower bound in terms of maximum degree for the minimum-density identifying codes of well-behaved infinite graphs. We prove that $d^*(H_2)=9/20$, $d^*(H_3)=6/13approx 0.4615$, $d^*(H_4)=7/16=0.4375$ and $d^*(H_5)=11/25=0:44$. We also prove that $H_2$ has a unique periodic identifying code with minimum density.
{"title":"Density of identifying codes of hexagonal grids with finite number of rows","authors":"Rudini Sampaio, Gabriel A. G. Sobral, Yoshiko Wakabayashi","doi":"10.1051/ro/2024046","DOIUrl":"https://doi.org/10.1051/ro/2024046","url":null,"abstract":"In a graph $G$, a set $Csubseteq V(G)$ is an identifying code if, for all vertices $v$ in $G$, the sets $N[v]cap C$ are all nonempty and pairwise distinct, where $N[v]$ denotes the closed neighbourhood of $v$. We focus on the minimum density of identifying codes of infinite hexagonal grids $H_k$ with $k$ rows, denoted by $d^*(H_k)$, and present optimal solutions for $kleq 5$. Using discharging method, we also prove a lower bound in terms of maximum degree for the minimum-density identifying codes of well-behaved infinite graphs. We prove that $d^*(H_2)=9/20$, $d^*(H_3)=6/13approx 0.4615$, $d^*(H_4)=7/16=0.4375$ and $d^*(H_5)=11/25=0:44$. We also prove that $H_2$ has a unique periodic identifying code with minimum density.","PeriodicalId":506995,"journal":{"name":"RAIRO - Operations Research","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139958472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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