The Factorization of Adjoint Polynomials of E^G（i）-class Graphs and Chromatically Equivalence Analysis

Let Sn be the star with n vertices, and let G be any connected graph with p vertices. We denote by ErGp(+i)(r-1) the graph obtained from Sr and rG by coinciding the i-th vertex of G with the vertex of degree r-1 of Sr, while the i-th vertex of each component of (r-1)G be adjacented to r-1 vertices of degree 1 of Sr, respectively. By applying the properties of adjoint polynomials, We prove that factorization theorem of adjoint polynomials of kinds of graphs ErGp(+i)(r-1) (r-1)K1 (1 ≤ i ≤ p). Furthermore, we obtain structure characteristics of chromatically equivalent graphs of their complements.

On Chromatically Equivalence of a Class of Graphs

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CIRCULAR CHROMATIC NUMBER AND MYCIELSKI GRAPHS

For a general graph G, M（G） denotes its Mycielski graph. This article gives a number of new sufficient conditions for G to have the circular chromatic number xc（M（G）） equals to the chromatic number x（M（G））, which have improved some best sufficient conditions published up to date.

EDGE-FACE CHROMATIC NUMBER OF 2-CONNECTED PLANE GRAPHS WITH HIGH MAXIMUM DEGREE

The edge-face chromatic number Xef （G） of a plane graph G is the least number of colors assigned to the edges and faces such that every adjacent or incident pair of them receives different colors. In this article, the authors prove that every 2-connected plane graph G with △（G）≥｜G｜ - 2≥9 has Xef（G） = △（G）.

RECURSIVE FORMULA FOR CALCULATING THE CHROMATIC POLYNOMIAL OF A GRAPH BY VERTEX DELETION

A new recursive vertex-deleting formula for the computation of the chromatic polynomial of a graph is obtained in this paper. This algorithm is not only a good tool for further studying chromatic polynomials but also the fastest among all the algorithms for the computation of chromatic polynomials.

Graph Description by the Chromatic Polynomials

借助于色彩的多项式，这份报纸为是的图 G 提供了一个必要、足够的条件一张独轮脚踏车图(定理 1 ) ，一张头等自行车图和一张二等自行车图(定理 2 ) 分别地。

ENTIRE CHROMATIC NUMBER AND Δ-MATCHING OF OUTERPLANE GRAPHS

Let G be an outerplane graph with maximum degree A and the entire chromatic number Xvef（G）. This paper proves that if △ ≥6, then △＋ 1≤Xvef（G）≤△＋ 2, and Xvef （G） = △＋ 1 if and only if G has a matching M consisting of some inner edges which covers all its vertices of maximum degree.

CHROMATIC NUMBER OF SQUARE OF MAXIMAL OUTERPLANAR GRAPHS

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.

Game Chromatic Number of Some Regular Graphs

Let G be a graph and k be a positive integer. We consider a game with two players Alice and Bob who alternate in coloring the vertices of G with a set of k colors. In every turn, one vertex will be chosen by one player. Alice’s goal is to color all vertices with the k colors, while Bob’s goal is to prevent her. The game chromatic number denoted by?χg(G), is the smallest k such that Alice has a winning strategy with k colors. In this paper, we determine the game chromatic number?χg of circulant graphs?Cn(1,2), , and generalized Petersen graphs GP(n,2), GP(n,3).

Chromatic Number of Graphs with Special Distance Sets-V

An integer distance graph is a graph G(Z,D) with the set of integers as vertex set and an edge joining two vertices u and?v if and only if ∣u - v∣D where D is a subset of the positive integers. It is known that x(G(Z,D) )=4 where P is a set of Prime numbers. So we can allocate the subsets D of P to four classes, accordingly as is 1 or 2 or 3 or 4. In this paper we have considered the open problem of characterizing class three and class four sets when the distance set D is not only a subset of primes P but also a special class of primes like Additive primes, Deletable primes, Wedderburn-Etherington Number primes, Euclid-Mullin sequence primes, Motzkin primes, Catalan primes, Schroder primes, Non-generous primes, Pell primes, Primeval primes, Primes of Binary Quadratic Form, Smarandache-Wellin primes, and Highly Cototient number primes. We also have indicated the membership of a number of special classes of prime numbers in class 2 category.

Geodetic Number and Geo-Chromatic Number of 2-Cartesian Product of Some Graphs

A set S ⊆ V (G) is called a geodetic set if every vertex of G lies on a shortest u-v path for some u, v ∈ S, the minimum cardinality among all geodetic sets is called geodetic number and is denoted by . A set C ⊆ V (G) is called a chromatic set if C contains all vertices of different colors in G, the minimum cardinality among all chromatic sets is called the chromatic number and is denoted by . A geo-chromatic set Sc ⊆ V (G) is both a geodetic set and a chromatic set. The geo-chromatic number of G is the minimum cardinality among all geo-chromatic sets of G. In this paper, we determine the geodetic number and the geo-chromatic number of 2-cartesian product of some standard graphs like complete graphs, cycles and paths.

The Equitable Total Chromatic Number of Some Join graphs

A proper total-coloring of graph G is said to be?equitable if the number of elements (vertices and edges) in any?two color classes differ by at most one, which the required?minimum number of colors is called the equitable total chromatic?number. In this paper, we prove some theorems on equitable?total coloring and derive the equitable total chromatic numbers?of Pm V?Sn, Pm V?Fn and Pm V Wn.

The Further Results of the Chromatic Uniqueness of Certain Bipartite Graphs K(m, n)-A

With its comprehensive application in network information engineering (e.g. dynamic spectrum allocation under different distance constraints) and in network combination optimization (e.g. safe storage of deleterious materials), the graphs ’ cloring theory and chromatic uniqueness theory have been the forward position of graph theory research. The later concerns the equivalent classification of graphs with their color polynomials and the determination of uniqueness of some equivalent classification under isomorphism. In this paper, by introducing the concept of chromatic normality and comparing the number of partitions of two chromatically equivalent graphs, a general numerical condition guarenteeing that bipartite graphs K(m, n)-A (AE(K(m, n)) and |A|≥2) is chromatically unique was obtained and a lot of chromatic uniqueness graphs of bipartite graphs K(m,n)-A were determined. The results obtained in this paper were general. And the results cover and extend the majority of the relevant results obtained within the world.

The Chromatic Uniqueness of Bipartite Graphs K(m,n)-A with |A|=2

The chromatically uniqueness of bipartite graphs K(m, n)- A(|A|=2) was studied. With comparing the numbers of partitions into r color classes of two chromatically equivalent graphs, one general numerical condition guaranteeing that K(m, n)-A(|A|=2) is chromatically unique were obtained. This covers and improves the former correlative results.

The Complete Chromatic Number of Maximal Outerplane Graphs

Let G be a maximal outerplane graph and X0(G) the complete chromatic number of G. This paper determines exactly X0(G) for △(G)≠5 and proves 6≤X0.(G)≤7 for △(G) = 5, where △(G) is the maximum degree of vertices of G.

On the Chromatic Number of (P5, C5, Cricket)-Free Graphs

For a graph G, let be the chromatic number of G. It is well-known that holds for any graph G with clique number . For a hereditary graph class , whether there exists a function f such that holds for every has been widely studied. Moreover, the form of minimum such an f is also concerned. A result of Schiermeyer shows that every -free graph G with clique number has . Chudnovsky and Sivaraman proved that every -free with clique number graph is -colorable. In this paper, for any -free graph G with clique number , we prove that . The main methods in the proof are set partition and induction.

THE TOTAL CHROMATIC NUMBER OF PSEUDO-OUTERPLANAR GRAPHS

A planar graph G is called a i pseudo outerplanar graph if there is a subset V 0V(G),|V 0|=i, such that G-V 0 is an outerplanar graph. In particular, when G-V 0 is a forest, G is called a i pseudo tree. In this paper, the following results are proved: (i) The conjecture on the total coloring is true for all 1 pseudo outerplanar graphs; (ii) χ t(G)=Δ(G)+1 for any 1 pseudo outerplanar graph G with Δ(G)6 and for any 1 pseudo tree G with Δ(G)3, where χ t(G) is the total chromatic number of a graph G .

一类仙人掌图的星边染色

图G的星边染色是指G的一个正常边染色,使得G中任一长为4的路和长为4的圈均不是2-边染色的.图G的星边色数χ’_(st)(G)表示图G有星边染色的最小颜色数.仙人掌图是一个连通图使得每个块是圈或者边.利用数学归纳法得到了一类仙人掌图C_(n)·C_(m)(n≥3,m≥3)的星边色数,从而推广已知结果.

F_(m)、P_(n)⊙F_(m)和C_(n)⊙F_(m)的r-hued染色研究

给定2个图G和H,它们的corona乘积图记为G⊙H,是将图G拷贝一份、图H拷贝|V(G)|份,图G的第i个顶点和图H的第i个拷贝份的每个顶点连边而得到的图.图G的(k,r)-染色是图G正常k-染色,使得度数为d的每个顶点的邻点至少染min{d,r}种不同的颜色.r-hued染色数是最小正整数k,使得图G具有(k,r)-染色,用χr(G)来表示.主要讨论F_(m),P_(n)⊙F_(m)和C_(n)⊙F_(m)的r-hued染色数.

一类仙人掌图的D(2)-点可区别全染色

用数学归纳法和组合分析法给出最大度为3的仙人掌图G T的D(2)-点可区别全染色,进而得到χ_(2vt)(G T)≤6.结果表明,D(β)-VDTC猜想对最大度为3的仙人掌图成立.