Pub Date : 2020-01-01DOI: 10.20429/tag.2020.070103
Kengo Enami, Seiya Negami
In this paper, we regard each edge of a connected graph G as a line segment having a unit length, and focus on not only the “vertices” but also any “point” lying along such a line segment. So we can define the distance between two points on G as the length of a shortest curve joining them along G . The beans function B G ( x ) of a connected graph G is defined as the maximum number of points on G such that any pair of points have distance at least x > 0. We shall show a recursive formula for B G ( x ) which enables us to determine the value of B G ( x ) for all x ≤ 1 by evaluating it only for 1 / 2 < x ≤ 1. As applications of this recursive formula, we shall propose an algorithm for computing B G ( x ) for a given value of x ≤ 1, and determine the beans functions of the complete graphs K n .
本文将连通图G的每条边视为一个单位长度的线段，不仅关注“顶点”，而且关注沿此线段的任何“点”。所以我们可以把G上两点之间的距离定义为G上连接两点的最短曲线的长度。连通图G的bean函数B G (x)定义为G上的最大点数，使得任何点对的距离至少为x >0 0。我们将给出一个G (x)的递归公式，它使我们能够通过仅在1 / 2 < x≤1时求值来确定G (x)在所有x≤1时的值。作为这个递归公式的应用，我们将提出一种计算给定值x≤1时的B G (x)的算法，并确定完全图K n的bean函数。
{"title":"Recursive Formulas for Beans Functions of Graphs","authors":"Kengo Enami, Seiya Negami","doi":"10.20429/tag.2020.070103","DOIUrl":"https://doi.org/10.20429/tag.2020.070103","url":null,"abstract":"In this paper, we regard each edge of a connected graph G as a line segment having a unit length, and focus on not only the “vertices” but also any “point” lying along such a line segment. So we can define the distance between two points on G as the length of a shortest curve joining them along G . The beans function B G ( x ) of a connected graph G is defined as the maximum number of points on G such that any pair of points have distance at least x > 0. We shall show a recursive formula for B G ( x ) which enables us to determine the value of B G ( x ) for all x ≤ 1 by evaluating it only for 1 / 2 < x ≤ 1. As applications of this recursive formula, we shall propose an algorithm for computing B G ( x ) for a given value of x ≤ 1, and determine the beans functions of the complete graphs K n .","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67545680","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}
Pub Date : 2020-01-01DOI: 10.20429/tag.2020.070107
L. Saha, A. R. Basunia
Radio labelling problem of graphs have its roots in communication problem known as Channel Assignment Problem . For a simple connected graph G = ( V ( G ) ; E ( G )), a radio labeling is a mapping f : V ( G ) → { 0 ; 1 ; 2 ; : : : } such that | f ( u ) − f ( v ) | ≥ diam( G )+ 1 − d ( u; v ) for each pair of distinct vertices u; v ∈ V ( G ), where diam(G) is the diameter of G and d ( u; v ) is the distance between u and v . A radio labeling f of a graph G is a radio graceful labeling of G if f ( V ( G )) = { 0 ; 1 ; : : : ; | V ( G ) | − 1 } . A graph for which a radio graceful labeling exists is called radio graceful . In this article, a necessary and sufficient condition for radio graceful graphs are presented. Also some consequences of radio graceful graphs are given in terms of some new graph parameters.
图的无线电标记问题根源于通信问题，即信道分配问题。对于简单连通图G = (V (G);E (G))，一个无线电标号是一个映射f: V (G)→{0;1;2;::}使得| f (u)−f (v) |≥diam(G)+ 1−d (u;V)对于每一对不同的顶点u;v∈v (G)，其中diam(G)为G的直径，d (u;V)是u和V之间的距离。如果f (V (G)) ={0，则图G的无线电标记f是G的无线电优美标记;1;:::;| v (g) |−1}。存在无线电优美标记的图称为无线电优美图。本文给出了无线电优美图的一个充分必要条件。同时给出了无线电优美图在一些新的图参数下的一些结果。
{"title":"Radio Graceful Labelling of Graphs","authors":"L. Saha, A. R. Basunia","doi":"10.20429/tag.2020.070107","DOIUrl":"https://doi.org/10.20429/tag.2020.070107","url":null,"abstract":"Radio labelling problem of graphs have its roots in communication problem known as Channel Assignment Problem . For a simple connected graph G = ( V ( G ) ; E ( G )), a radio labeling is a mapping f : V ( G ) → { 0 ; 1 ; 2 ; : : : } such that | f ( u ) − f ( v ) | ≥ diam( G )+ 1 − d ( u; v ) for each pair of distinct vertices u; v ∈ V ( G ), where diam(G) is the diameter of G and d ( u; v ) is the distance between u and v . A radio labeling f of a graph G is a radio graceful labeling of G if f ( V ( G )) = { 0 ; 1 ; : : : ; | V ( G ) | − 1 } . A graph for which a radio graceful labeling exists is called radio graceful . In this article, a necessary and sufficient condition for radio graceful graphs are presented. Also some consequences of radio graceful graphs are given in terms of some new graph parameters.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67545785","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}
Pub Date : 2019-09-16DOI: 10.20429/tag.2019.060202
Huifen Ge, Tianlong Ma, Miaolin Wu, Yuzhi Xiao
Let F be a subset of edges and vertices of a graph G. If G − F has no fractional perfect matching, then F is a fractional strong matching preclusion set of G. The fractional strong matching preclusion number is the cardinality of a minimum fractional strong matching preclusion set. In this paper, we mainly study the fractional strong matching preclusion problem for two variants of hypercubes, the multiply twisted cube and the locally twisted cube, which are two of the most popular interconnection networks. In addition, we classify all the optimal fractional strong matching preclusion set of each.
{"title":"Fractional strong matching preclusion for two variants of hypercubes","authors":"Huifen Ge, Tianlong Ma, Miaolin Wu, Yuzhi Xiao","doi":"10.20429/tag.2019.060202","DOIUrl":"https://doi.org/10.20429/tag.2019.060202","url":null,"abstract":"Let F be a subset of edges and vertices of a graph G. If G − F has no fractional perfect matching, then F is a fractional strong matching preclusion set of G. The fractional strong matching preclusion number is the cardinality of a minimum fractional strong matching preclusion set. In this paper, we mainly study the fractional strong matching preclusion problem for two variants of hypercubes, the multiply twisted cube and the locally twisted cube, which are two of the most popular interconnection networks. In addition, we classify all the optimal fractional strong matching preclusion set of each.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48960553","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}
Pub Date : 2019-08-30DOI: 10.20429/tag.2019.060201
Colton Magnant
The study of graph Ramsey numbers within restricted colorings, in particular forbidding a rainbow triangle, has recently been blossoming under the name GallaiRamsey numbers. In this work, we extend the main structural tool from rainbow triangle free colorings of complete graphs to rainbow Berge triangle free colorings of hypergraphs. In doing so, some other concepts and results are also translated from graphs to hypergraphs.
{"title":"Colored complete hypergraphs containing no rainbow Berge triangles","authors":"Colton Magnant","doi":"10.20429/tag.2019.060201","DOIUrl":"https://doi.org/10.20429/tag.2019.060201","url":null,"abstract":"The study of graph Ramsey numbers within restricted colorings, in particular forbidding a rainbow triangle, has recently been blossoming under the name GallaiRamsey numbers. In this work, we extend the main structural tool from rainbow triangle free colorings of complete graphs to rainbow Berge triangle free colorings of hypergraphs. In doing so, some other concepts and results are also translated from graphs to hypergraphs.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49472985","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}
Pub Date : 2019-05-01DOI: 10.20429/TAG.2019.060105
Ajay Arora, E. Cheng, Christopher Melekian
The matching preclusion number of a graph with an even number of vertices is the minimum number of edges whose deletion results in a graph with no perfect matchings. In this paper we determine the matching preclusion number for the generalized Petersen graph P (n, k) and classify the optimal sets.
{"title":"Matching Preclusion of the Generalized Petersen Graph","authors":"Ajay Arora, E. Cheng, Christopher Melekian","doi":"10.20429/TAG.2019.060105","DOIUrl":"https://doi.org/10.20429/TAG.2019.060105","url":null,"abstract":"The matching preclusion number of a graph with an even number of vertices is the minimum number of edges whose deletion results in a graph with no perfect matchings. In this paper we determine the matching preclusion number for the generalized Petersen graph P (n, k) and classify the optimal sets.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46345676","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}
Pub Date : 2019-04-25DOI: 10.20429/TAG.2019.060104
Christian Barrientos, S. Minion
Among difference vertex labelings of graphs, α-labelings are the most restrictive one. A graph is an α-graph if it admits an α-labeling. In this work, we study a new alternative to construct α-graphs using, the well-known, series-parallel operations on smaller α-graphs. As an application of the series operation, we show that all members of a subfamily of all trees with maximum degree 4, obtained using vertex amalgamation of copies of the path P11, are α-graphs. We also show that the one-point union of up to four copies of Kn,n is an α-graph. In addition we prove that any α-graph of order m and size n is an induced subgraph of a graph of order m+ 2 and size m+ n. Furthermore, we prove that the Cartesian product of the bipartite graph K2,n and the path Pm is an α-graph.
{"title":"Series-Parallel Operations with Alpha-Graphs","authors":"Christian Barrientos, S. Minion","doi":"10.20429/TAG.2019.060104","DOIUrl":"https://doi.org/10.20429/TAG.2019.060104","url":null,"abstract":"Among difference vertex labelings of graphs, α-labelings are the most restrictive one. A graph is an α-graph if it admits an α-labeling. In this work, we study a new alternative to construct α-graphs using, the well-known, series-parallel operations on smaller α-graphs. As an application of the series operation, we show that all members of a subfamily of all trees with maximum degree 4, obtained using vertex amalgamation of copies of the path P11, are α-graphs. We also show that the one-point union of up to four copies of Kn,n is an α-graph. In addition we prove that any α-graph of order m and size n is an induced subgraph of a graph of order m+ 2 and size m+ n. Furthermore, we prove that the Cartesian product of the bipartite graph K2,n and the path Pm is an α-graph.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43418374","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}
Pub Date : 2019-01-27DOI: 10.20429/TAG.2019.060101
Y. Caro, Z. Tuza
We say that a subgraph $F$ of a graph $G$ is singular if the degrees $d_G(v)$ are all equal or all distinct for the vertices $vin V(F)$. The singular Ramsey number Rs$(F)$ is the smallest positive integer $n$ such that, for every $mgeq n$, in every edge 2-coloring of $K_m$, at least one of the color classes contains $F$ as a singular subgraph. In a similar flavor, the singular Tur'an number Ts$(n,F)$ is defined as the maximum number of edges in a graph of order $n$, which does not contain $F$ as a singular subgraph. In this paper we initiate the study of these extremal problems. We develop methods to estimate Rs$(F)$ and Ts$(n,F)$, present tight asymptotic bounds and exact results.
{"title":"Singular Ramsey and Turán numbers","authors":"Y. Caro, Z. Tuza","doi":"10.20429/TAG.2019.060101","DOIUrl":"https://doi.org/10.20429/TAG.2019.060101","url":null,"abstract":"We say that a subgraph $F$ of a graph $G$ is singular if the degrees $d_G(v)$ are all equal or all distinct for the vertices $vin V(F)$. The singular Ramsey number Rs$(F)$ is the smallest positive integer $n$ such that, for every $mgeq n$, in every edge 2-coloring of $K_m$, at least one of the color classes contains $F$ as a singular subgraph. In a similar flavor, the singular Tur'an number Ts$(n,F)$ is defined as the maximum number of edges in a graph of order $n$, which does not contain $F$ as a singular subgraph. In this paper we initiate the study of these extremal problems. We develop methods to estimate Rs$(F)$ and Ts$(n,F)$, present tight asymptotic bounds and exact results.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47181404","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}
Pub Date : 2019-01-01DOI: 10.20429/tag.2019.060103
Xia Wang, Tianlong Ma, Chengfu Ye, Yuzhi Xiao, F. Wang
The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. As a generalization, Liu and Liu [17] recently introduced the concept of fractional matching preclusion number. The fractional matching preclusion number (FMP number) of G, denoted by fmp(G), is the minimum number of edges whose deletion leaves the resulting graph without a fractional perfect matching. The fractional strong matching preclusion number (FSMP number) of G, denoted by fsmp(G), is the minimum number of vertices and edges whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we study the fractional matching preclusion number and the fractional strong matching preclusion number for butterfly network, augmented butterfly network and enhanced butterfly network.
{"title":"Fractional matching preclusion for butterfly derived networks","authors":"Xia Wang, Tianlong Ma, Chengfu Ye, Yuzhi Xiao, F. Wang","doi":"10.20429/tag.2019.060103","DOIUrl":"https://doi.org/10.20429/tag.2019.060103","url":null,"abstract":"The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. As a generalization, Liu and Liu [17] recently introduced the concept of fractional matching preclusion number. The fractional matching preclusion number (FMP number) of G, denoted by fmp(G), is the minimum number of edges whose deletion leaves the resulting graph without a fractional perfect matching. The fractional strong matching preclusion number (FSMP number) of G, denoted by fsmp(G), is the minimum number of vertices and edges whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we study the fractional matching preclusion number and the fractional strong matching preclusion number for butterfly network, augmented butterfly network and enhanced butterfly network.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67545248","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}
Pub Date : 2019-01-01DOI: 10.20429/tag.2019.060203
F. Motialah, M. H. S. Haghighi
A sun SGn is a graph of order 2n consisting of a cycle Cn, n ≥ 3, to each vertex of it a pendant edge is attached. In this paper, we prove that unbalanced signed sun graphs are determined by their Laplacian spectra. Also we show that a balanced signed sun graph is determined by its Laplacian spectrum if and only if n is odd.
{"title":"Laplacian Spectral Characterization of Signed Sun Graphs","authors":"F. Motialah, M. H. S. Haghighi","doi":"10.20429/tag.2019.060203","DOIUrl":"https://doi.org/10.20429/tag.2019.060203","url":null,"abstract":"A sun SGn is a graph of order 2n consisting of a cycle Cn, n ≥ 3, to each vertex of it a pendant edge is attached. In this paper, we prove that unbalanced signed sun graphs are determined by their Laplacian spectra. Also we show that a balanced signed sun graph is determined by its Laplacian spectrum if and only if n is odd.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67545581","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}
Pub Date : 2019-01-01DOI: 10.20429/tag.2019.060204
Kai Wang
We present an algorithm to test whether a given graphical degree sequence is forcibly biconnected. The worst case time complexity of the algorithm is shown to be exponential but it is still much better than the previous basic algorithm for this problem. We show through experimental evaluations that the algorithm is efficient on average. We also adapt the classic algorithm of Ruskey et al. and that of Barnes and Savage to obtain some enumerative results about forcibly biconnected graphical degree sequences of given length n and forcibly biconnected graphical partitions of given even integer n . Based on these enumerative results we make some conjectures such as: when n is large, (1) the proportion of forcibly biconnected graphical degree sequences of length n among all zero-free graphical degree sequences of length n is asymptotically a constant C (0 < C < 1); (2) the proportion of forcibly biconnected graphical partitions of even n among all forcibly connected graphical partitions of n is asymptotically 0.
提出了一种检验给定图形度序列是否强制双连通的算法。在最坏情况下，该算法的时间复杂度为指数级，但仍然比以前的基本算法好得多。实验结果表明，该算法总体上是有效的。我们还采用了Ruskey等人的经典算法和Barnes和Savage的经典算法，得到了关于给定长度n的强制双连通图度序列和给定偶数n的强制双连通图分区的一些枚举结果。基于这些枚举结果，我们做出了一些猜想，如:当n较大时，(1)长度为n的强制双连通图度序列占所有长度为n的零自由图度序列的比例渐近为常数C (0 < C < 1);(2)偶数n的强制双连通图分区占所有强制连通图分区的比例为
{"title":"Forcibly-biconnected Graphical Degree Sequences: Decision Algorithms and Enumerative Results","authors":"Kai Wang","doi":"10.20429/tag.2019.060204","DOIUrl":"https://doi.org/10.20429/tag.2019.060204","url":null,"abstract":"We present an algorithm to test whether a given graphical degree sequence is forcibly biconnected. The worst case time complexity of the algorithm is shown to be exponential but it is still much better than the previous basic algorithm for this problem. We show through experimental evaluations that the algorithm is efficient on average. We also adapt the classic algorithm of Ruskey et al. and that of Barnes and Savage to obtain some enumerative results about forcibly biconnected graphical degree sequences of given length n and forcibly biconnected graphical partitions of given even integer n . Based on these enumerative results we make some conjectures such as: when n is large, (1) the proportion of forcibly biconnected graphical degree sequences of length n among all zero-free graphical degree sequences of length n is asymptotically a constant C (0 < C < 1); (2) the proportion of forcibly biconnected graphical partitions of even n among all forcibly connected graphical partitions of n is asymptotically 0.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67545606","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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