Let x（G^2） denote the chromatic number of the square of a maximal outerplanar graph G and Q denote a maximal outerplanar graph obtained by adding three chords y1 y3, y3y5, y5y1 to a 6-cycle y1y2…y6y1. In this paper...Let x（G^2） denote the chromatic number of the square of a maximal outerplanar graph G and Q denote a maximal outerplanar graph obtained by adding three chords y1 y3, y3y5, y5y1 to a 6-cycle y1y2…y6y1. In this paper, it is proved that △ ＋ 1 ≤ x（G^2） ≤△ ＋ 2, and x（G^2） = A ＋ 2 if and only if G is Q, where A represents the maximum degree of G.展开更多
Let G be a simple graph with no isolated edge. An/-total coloring of a graphG is a mapping Ф ： V（G） U E（G） → （1, 2,…… , k） such that no adjacent vertices receive thesame color and no adjacent edges receive ...Let G be a simple graph with no isolated edge. An/-total coloring of a graphG is a mapping Ф ： V（G） U E（G） → （1, 2,…… , k） such that no adjacent vertices receive thesame color and no adjacent edges receive the same color. An/-total coloring of a graph G issaid to be adjacent vertex distinguishing if for any pair of adjacent vertices u and v of G, wehave CФ（u） ≠ CФ（v）, where CФ（u） denotes the set of colors of u and its incident edges. Theminimum number of colors required for an adjacent vertex distinguishing I-total coloring of GG is called the adjacent vertex distinguishing I-total chromatic number, denoted by Xat（G）.In this paper, we characterize the adjacent vertex distinguishing I-total chromatic numberof outerplanar graphs.展开更多
Catalan number is an important class of combinatorial numbers. The maximal outerplanar graphs are important in graph theory. In this paper some formulas to enumerate the numbers of maximal outerplanar graphs by means ...Catalan number is an important class of combinatorial numbers. The maximal outerplanar graphs are important in graph theory. In this paper some formulas to enumerate the numbers of maximal outerplanar graphs by means of the compressing graph and group theory method are given first. Then the relationships between Catalan numbers and the numbers of labeled and unlabeled maximal outerplanar graphs are presented. The computed results verified these formulas.展开更多
All maps we deal with in this note are rooted planar ones, which, in brief, are called maps. Most of terminologies here come from Refs. [1—3]. The purpose of this note is to determine the asymptotic enumerations base...All maps we deal with in this note are rooted planar ones, which, in brief, are called maps. Most of terminologies here come from Refs. [1—3]. The purpose of this note is to determine the asymptotic enumerations based on the enumerations in [1, 2]. Bender and Richmond gave a survey of the asymptotic enumeration. New asymptotic enumerations here can be considered as a complement and improvement of the survey.展开更多
An edge coloring total k-labeling is a labeling of the vertices and the edges of a graph G with labels {1,2,..., k} such that the weights of the edges define a proper edge coloring of G. Here the weight of an edge is ...An edge coloring total k-labeling is a labeling of the vertices and the edges of a graph G with labels {1,2,..., k} such that the weights of the edges define a proper edge coloring of G. Here the weight of an edge is the sum of its label and the labels of its two end vertices. This concept was introduce by Brandt et al. They defined Xt＇（G） to be the smallest integer k for which G has an edge coloring total k-labeling and proposed a question： Is there a constant K with X^t（G） ≤△（G）＋1/2 K for all graphs G of maximum degree A（G）？ In this paper, we give a positive answer for outerplanar graphs ≤△（G）＋1/2 by showing that X＇t（G） ≤△（G）＋1/2 for each outerplanar graph G with maximum degree A（G）.展开更多
A strong k-edge-coloring of a graph G is an assignment of k colors to the edges of G in such a way that any two edges meeting at a common vertex, or being adjacent to the same edge of G, axe assigned different colors....A strong k-edge-coloring of a graph G is an assignment of k colors to the edges of G in such a way that any two edges meeting at a common vertex, or being adjacent to the same edge of G, axe assigned different colors. The strong chromatic index of G is the smallest integer k for which G has a strong k-edge-coloring. In this paper, we have shown that the strong chromatic index is no larger than 6 for outerplanax graphs with maximum degree 3.展开更多
The spectral radius is an important parameter of a graph related to networks. A method for estimating the spectral radius of each spanning subgraph is used to prove that the spectral radius of a Hamiltonian planar g...The spectral radius is an important parameter of a graph related to networks. A method for estimating the spectral radius of each spanning subgraph is used to prove that the spectral radius of a Hamiltonian planar graph of order n≥4 is less than or equal to 2+3n-11 and the spectral radius of the outerplanar graph of order n≥6 is less than or equal to 22+n-5, which are improvements over previous results. A direction for further study is then suggested.展开更多
Let G be an outerplanar graph with maximum degree △. Let χ（G^2） and A（G） denote the chromatic number of the square and the L（2, 1）-labelling number of G, respectively. In this paper we prove the following resu...Let G be an outerplanar graph with maximum degree △. Let χ（G^2） and A（G） denote the chromatic number of the square and the L（2, 1）-labelling number of G, respectively. In this paper we prove the following results： （1） χ（G^2） = 7 if △= 6; （2） λ（G） ≤ △ ＋5 if △ ≥ 4, and ）,（G）≤ 7 if △ = 3; and （3） there is an outerplanar graph G with △ = 4 such that ）λ（G） = 7. These improve some known results on the distance two labelling of outerplanar graphs.展开更多
In this paper, the number of combinatorially distinct rooted nonseparable outerplanar maps withm edges and the valency of the root-face being n is found to be(m-1)! (m-2) !:(n-1)!(n-2)! (m-n)!(m-n+1)!and, the number o...In this paper, the number of combinatorially distinct rooted nonseparable outerplanar maps withm edges and the valency of the root-face being n is found to be(m-1)! (m-2) !:(n-1)!(n-2)! (m-n)!(m-n+1)!and, the number of rooted nonseparable outerplanar maps with m edges is also determined to be(2m-2)!:(m-1)!m!,which is just the number of distinct rooted plane trees with m-1 edges.展开更多
In this paper we give an enumeration formula of the outerplanar graphs by means of graph compression, group theory and combinatorial numbers. Some simple examples are exhibited for illustrating the method. The computa...In this paper we give an enumeration formula of the outerplanar graphs by means of graph compression, group theory and combinatorial numbers. Some simple examples are exhibited for illustrating the method. The computational results are shown in the table at the end of this paper.展开更多
The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known vertex covering and dominating set problems in graph theory. In t...The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known vertex covering and dominating set problems in graph theory. In this paper, it was shown that the power domination number of an outerplanar graph with the diameter two or a 2-connected outerplanar graph with the diameter three is precisely one. Upper bounds on the power domination number for a general planar graph with the diameter two or three were determined as an immediate consequences of results proven by Dorfling, et al. Also, an infinite family of outerplanar graphs with the diameter four having arbitrarily large power domination numbers were given.展开更多
文摘Let x（G^2） denote the chromatic number of the square of a maximal outerplanar graph G and Q denote a maximal outerplanar graph obtained by adding three chords y1 y3, y3y5, y5y1 to a 6-cycle y1y2…y6y1. In this paper, it is proved that △ ＋ 1 ≤ x（G^2） ≤△ ＋ 2, and x（G^2） = A ＋ 2 if and only if G is Q, where A represents the maximum degree of G.
基金Supported by the National Natural Science Foundation of China(61163037,61163054,61363060)
文摘Let G be a simple graph with no isolated edge. An/-total coloring of a graphG is a mapping Ф ： V（G） U E（G） → （1, 2,…… , k） such that no adjacent vertices receive thesame color and no adjacent edges receive the same color. An/-total coloring of a graph G issaid to be adjacent vertex distinguishing if for any pair of adjacent vertices u and v of G, wehave CФ（u） ≠ CФ（v）, where CФ（u） denotes the set of colors of u and its incident edges. Theminimum number of colors required for an adjacent vertex distinguishing I-total coloring of GG is called the adjacent vertex distinguishing I-total chromatic number, denoted by Xat（G）.In this paper, we characterize the adjacent vertex distinguishing I-total chromatic numberof outerplanar graphs.
文摘Catalan number is an important class of combinatorial numbers. The maximal outerplanar graphs are important in graph theory. In this paper some formulas to enumerate the numbers of maximal outerplanar graphs by means of the compressing graph and group theory method are given first. Then the relationships between Catalan numbers and the numbers of labeled and unlabeled maximal outerplanar graphs are presented. The computed results verified these formulas.
基金Project supported by the National Natural Science Foundation of China.
文摘All maps we deal with in this note are rooted planar ones, which, in brief, are called maps. Most of terminologies here come from Refs. [1—3]. The purpose of this note is to determine the asymptotic enumerations based on the enumerations in [1, 2]. Bender and Richmond gave a survey of the asymptotic enumeration. New asymptotic enumerations here can be considered as a complement and improvement of the survey.
基金Supported by National Natural Science Foundation of China(Grant Nos.61070230 and 11101243)Doctoral Fund of Ministry of Education of China(Grant No.20100131120017)the Scientific Research Foundation for the Excellent Middle-Aged and Young Scientists of Shandong Province(Grant No.BS2012SF016)
文摘An edge coloring total k-labeling is a labeling of the vertices and the edges of a graph G with labels {1,2,..., k} such that the weights of the edges define a proper edge coloring of G. Here the weight of an edge is the sum of its label and the labels of its two end vertices. This concept was introduce by Brandt et al. They defined Xt＇（G） to be the smallest integer k for which G has an edge coloring total k-labeling and proposed a question： Is there a constant K with X^t（G） ≤△（G）＋1/2 K for all graphs G of maximum degree A（G）？ In this paper, we give a positive answer for outerplanar graphs ≤△（G）＋1/2 by showing that X＇t（G） ≤△（G）＋1/2 for each outerplanar graph G with maximum degree A（G）.
基金Supported by the National Natural Science Foundation of China under Grant No.11501050the Fundamental Research Funds for the Central Universities under Grant No.310812151003
文摘A strong k-edge-coloring of a graph G is an assignment of k colors to the edges of G in such a way that any two edges meeting at a common vertex, or being adjacent to the same edge of G, axe assigned different colors. The strong chromatic index of G is the smallest integer k for which G has a strong k-edge-coloring. In this paper, we have shown that the strong chromatic index is no larger than 6 for outerplanax graphs with maximum degree 3.
基金the National Natural Science Foundationof China (No.196 710 5 0 )
文摘The spectral radius is an important parameter of a graph related to networks. A method for estimating the spectral radius of each spanning subgraph is used to prove that the spectral radius of a Hamiltonian planar graph of order n≥4 is less than or equal to 2+3n-11 and the spectral radius of the outerplanar graph of order n≥6 is less than or equal to 22+n-5, which are improvements over previous results. A direction for further study is then suggested.
基金Supported by the National Natural Science Foundation of China(No.10771197)
文摘Let G be an outerplanar graph with maximum degree △. Let χ（G^2） and A（G） denote the chromatic number of the square and the L（2, 1）-labelling number of G, respectively. In this paper we prove the following results： （1） χ（G^2） = 7 if △= 6; （2） λ（G） ≤ △ ＋5 if △ ≥ 4, and ）,（G）≤ 7 if △ = 3; and （3） there is an outerplanar graph G with △ = 4 such that ）λ（G） = 7. These improve some known results on the distance two labelling of outerplanar graphs.
基金The Project Supported by National Natural Science Foundation of China
文摘In this paper, the number of combinatorially distinct rooted nonseparable outerplanar maps withm edges and the valency of the root-face being n is found to be(m-1)! (m-2) !:(n-1)!(n-2)! (m-n)!(m-n+1)!and, the number of rooted nonseparable outerplanar maps with m edges is also determined to be(2m-2)!:(m-1)!m!,which is just the number of distinct rooted plane trees with m-1 edges.
文摘In this paper we give an enumeration formula of the outerplanar graphs by means of graph compression, group theory and combinatorial numbers. Some simple examples are exhibited for illustrating the method. The computational results are shown in the table at the end of this paper.
基金Project supporte（t by the National Natural Science Foundation of China （Grant No.10571117）, and the Youth Science Foundation of Shanghai Municipal Commission of Education （Grant No.01QN6262）
文摘The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known vertex covering and dominating set problems in graph theory. In this paper, it was shown that the power domination number of an outerplanar graph with the diameter two or a 2-connected outerplanar graph with the diameter three is precisely one. Upper bounds on the power domination number for a general planar graph with the diameter two or three were determined as an immediate consequences of results proven by Dorfling, et al. Also, an infinite family of outerplanar graphs with the diameter four having arbitrarily large power domination numbers were given.
