{"title":"On the permutation cycle structures for almost Moore digraphs of degrees 4 and 5","authors":"","doi":"10.47443/dml.2023.142","DOIUrl":"https://doi.org/10.47443/dml.2023.142","url":null,"abstract":"","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":"44 16","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138952517","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}
{"title":"Atom-bond sum-connectivity index of line graphs","authors":"","doi":"10.47443/dml.2023.197","DOIUrl":"https://doi.org/10.47443/dml.2023.197","url":null,"abstract":"","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":"48 5","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138953157","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}
For trees with given number of vertices n and maximum degree ∆ , we present lower bounds on the general sum-connectivity index χ a if a > 0 and 3 ≤ ∆ ≤ n − 1 , and an upper bound on the general Randi´c index R a if − 0 . 283 ≤ a < 0 and 3 ≤ ∆ ≤ (cid:98) n − 1 2 (cid:99) . All the extremal trees for our bounds are given.
对于具有给定顶点数 n 和最大度数 ∆ 的树，如果 a > 0 且 3 ≤ ∆ ≤ n - 1，我们给出了一般和连接性指数 χ a 的下限；如果 - 0 . 283 ≤ a < 0 且 3 ≤ ∆ ≤ (cid:98) n - 1 2 (cid:99) ，我们给出了一般 Randi´c 指数 R a 的上限。我们给出了边界的所有极值树。
{"title":"General sum-connectivity index and general Randic index of trees with given maximum degree","authors":"Elize Swartz, Tom´aˇs Vetr´ık","doi":"10.47443/dml.2023.140","DOIUrl":"https://doi.org/10.47443/dml.2023.140","url":null,"abstract":"For trees with given number of vertices n and maximum degree ∆ , we present lower bounds on the general sum-connectivity index χ a if a > 0 and 3 ≤ ∆ ≤ n − 1 , and an upper bound on the general Randi´c index R a if − 0 . 283 ≤ a < 0 and 3 ≤ ∆ ≤ (cid:98) n − 1 2 (cid:99) . All the extremal trees for our bounds are given.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":"24 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139010369","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}
{"title":"Extremal trees with fixed degree sequence for $sigma$-irregularity","authors":"","doi":"10.47443/dml.2023.106","DOIUrl":"https://doi.org/10.47443/dml.2023.106","url":null,"abstract":"","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":"359 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139233244","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 total vertex irregularity strength tvs( G ) of a simple graph G ( V, E ) is the smallest positive integer k so that there exists a function ϕ : V ∪ E → [1 , k ] provided that all vertex-weights are distinct, where a vertex-weight is the sum of labels of a vertex and all of its incident edges. In the paper [Nurdin, E. T. Baskoro, A. N. M. Salman, N. N. Gaos, Discrete Math. 310 (2010) 3043–3048], two conjectures regarding the total vertex irregularity strength of trees and general graphs were posed as follows: (i) for every tree T , tvs( T ) = max {(cid:100) ( n 1 + 1) / 2 (cid:101) , (cid:100) ( n 1 + n 2 + 1) / 3 (cid:101) , (cid:100) ( n 1 + n 2 + n 3 + 1) / 4 (cid:101)} , and (ii) for every graph G with minimum degree δ and maximum degree ∆ , tvs( G ) = max {(cid:100) ( δ + (cid:80) i j =1 n j ) / ( i + 1) (cid:101) : i ∈ [ δ, ∆] } , where n j denotes the number of vertices of degree j . In this paper, we disprove both of these conjectures by giving infinite families of counterexamples.
简单图 G ( V, E ) 的总顶点不规则强度 tvs( G ) 是存在一个函数 ϕ : V ∪ E → [1 , k ] 的最小正整数，条件是所有顶点权重都是不同的，其中顶点权重是一个顶点及其所有入射边的标签之和。在论文 [Nurdin, E. T. Baskoro, A. N. M. Salman, N. N. Gaos, Discrete Math.310 (2010) 3043-3048] 中，提出了关于树和一般图的总顶点不规则强度的两个猜想如下：(i) 对于每一棵树 T , tvs( T ) = max {(cid:100) ( n 1 + 1) / 2 (cid:101) , (cid:100) ( n 1 + n 2 + 1) / 3 (cid:101) , (cid:100) ( n 1 + n 2 + n 3 + 1) / 4 (cid：(ii) 对于具有最小度 δ 和最大度 ∆ 的每个图 G，tvs( G ) = max {(cid:100) ( δ + (cid:80) i j =1 n j ) / ( i + 1) (cid:101) : i∈ [ δ, ∆] } ，其中 n j 表示图 G 的个数。｝其中 n j 表示度数为 j 的顶点数。在本文中，我们通过给出有限的反例族来反证这两个猜想。
{"title":"Counterexamples to the total vertex irregularity strength’s conjectures","authors":"Faisal Susanto, R. Simanjuntak, E. Baskoro","doi":"10.47443/dml.2023.111","DOIUrl":"https://doi.org/10.47443/dml.2023.111","url":null,"abstract":"The total vertex irregularity strength tvs( G ) of a simple graph G ( V, E ) is the smallest positive integer k so that there exists a function ϕ : V ∪ E → [1 , k ] provided that all vertex-weights are distinct, where a vertex-weight is the sum of labels of a vertex and all of its incident edges. In the paper [Nurdin, E. T. Baskoro, A. N. M. Salman, N. N. Gaos, Discrete Math. 310 (2010) 3043–3048], two conjectures regarding the total vertex irregularity strength of trees and general graphs were posed as follows: (i) for every tree T , tvs( T ) = max {(cid:100) ( n 1 + 1) / 2 (cid:101) , (cid:100) ( n 1 + n 2 + 1) / 3 (cid:101) , (cid:100) ( n 1 + n 2 + n 3 + 1) / 4 (cid:101)} , and (ii) for every graph G with minimum degree δ and maximum degree ∆ , tvs( G ) = max {(cid:100) ( δ + (cid:80) i j =1 n j ) / ( i + 1) (cid:101) : i ∈ [ δ, ∆] } , where n j denotes the number of vertices of degree j . In this paper, we disprove both of these conjectures by giving infinite families of counterexamples.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":"11 5 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139234445","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 class of permutations avoiding the complement of an arbitrary regular permutation group is described.
{"title":"Permutations avoiding the complement of a regular permutation group","authors":"","doi":"10.47443/dml.2023.141","DOIUrl":"https://doi.org/10.47443/dml.2023.141","url":null,"abstract":"The class of permutations avoiding the complement of an arbitrary regular permutation group is described.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":"29 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136346313","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}
Let G be a graph of order n . A spanning subgraph F of G is said to be a P ≥ k -factor of G if every component of F is a path with at least k vertices, where k ≥ 2 . In this paper, we introduce the concept of an ID-P ≥ k -factor critical graph; a graph G is said to be an ID-P ≥ k -factor critical graph if for any independent set I of G , G − I admits a P ≥ k -factor. We prove that a graph G of a given order is an ID-P ≥ 2 -factor critical graph if its binding number is at least 2 . We also prove that a graph G of a fixed order is an ID-P ≥ 3 -factor critical graph if its binding number is at least 94 . Furthermore, we show that the obtained results are the best possible in some sense.
{"title":"Some results about ID-path-factor critical graphs","authors":"","doi":"10.47443/dml.2023.131","DOIUrl":"https://doi.org/10.47443/dml.2023.131","url":null,"abstract":"Let G be a graph of order n . A spanning subgraph F of G is said to be a P ≥ k -factor of G if every component of F is a path with at least k vertices, where k ≥ 2 . In this paper, we introduce the concept of an ID-P ≥ k -factor critical graph; a graph G is said to be an ID-P ≥ k -factor critical graph if for any independent set I of G , G − I admits a P ≥ k -factor. We prove that a graph G of a given order is an ID-P ≥ 2 -factor critical graph if its binding number is at least 2 . We also prove that a graph G of a fixed order is an ID-P ≥ 3 -factor critical graph if its binding number is at least 94 . Furthermore, we show that the obtained results are the best possible in some sense.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136348549","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 this paper, the t -color star-critical Gallai-Ramsey number for a path of order 5 is determined. It is proved that t +1 edges are both necessary and sufficient to add between a vertex and a critical coloring for the t -color Gallai-Ramsey number for P 5 in order to guarantee the existence of a monochromatic subgraph isomorphic to P 5 . The proof depends on a well-known structural result for Gallai colorings as well as a general lower bound due to Faudree, Gould, Jacobson, and Magnant.
{"title":"The multicolor star-critical Gallai-Ramsey number for a path of order 5","authors":"","doi":"10.47443/dml.2023.115","DOIUrl":"https://doi.org/10.47443/dml.2023.115","url":null,"abstract":"In this paper, the t -color star-critical Gallai-Ramsey number for a path of order 5 is determined. It is proved that t +1 edges are both necessary and sufficient to add between a vertex and a critical coloring for the t -color Gallai-Ramsey number for P 5 in order to guarantee the existence of a monochromatic subgraph isomorphic to P 5 . The proof depends on a well-known structural result for Gallai colorings as well as a general lower bound due to Faudree, Gould, Jacobson, and Magnant.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136032906","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}
A zonal labeling of a plane graph G is an assignment of the two nonzero elements of the ring Z 3 of integers modulo 3 to the vertices of G such that the sum of the labels of the vertices on the boundary of each region of G is the zero element of Z 3 . A plane graph possessing such a labeling is a zonal graph. If there is at most one exception, then the labeling is inner zonal and the graph is inner zonal. In 2019, Chartrand, Egan, and Zhang proved that showing the existence of zonal labelings in all cubic maps is equivalent to giving a proof of the Four Color Theorem. It is shown that every inner zonal cubic map is zonal, thereby establishing an improvement of the 2019 result. It is also shown that (i) while certain 2 -connected plane graphs of maximum degree 3 may not be zonal, they must be inner zonal and (ii) no connected cubic plane graph with bridges can be inner zonal.
{"title":"On zonal and inner zonal labelings of plane graphs of maximum degree 3","authors":"Andrew Bowling, Ping Zhang","doi":"10.47443/dml.2023.083","DOIUrl":"https://doi.org/10.47443/dml.2023.083","url":null,"abstract":"A zonal labeling of a plane graph G is an assignment of the two nonzero elements of the ring Z 3 of integers modulo 3 to the vertices of G such that the sum of the labels of the vertices on the boundary of each region of G is the zero element of Z 3 . A plane graph possessing such a labeling is a zonal graph. If there is at most one exception, then the labeling is inner zonal and the graph is inner zonal. In 2019, Chartrand, Egan, and Zhang proved that showing the existence of zonal labelings in all cubic maps is equivalent to giving a proof of the Four Color Theorem. It is shown that every inner zonal cubic map is zonal, thereby establishing an improvement of the 2019 result. It is also shown that (i) while certain 2 -connected plane graphs of maximum degree 3 may not be zonal, they must be inner zonal and (ii) no connected cubic plane graph with bridges can be inner zonal.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48042922","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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