On the Line Graph of the Complement Graph for the Ring of Gaussian Integers Modulo n

The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally and locally connected is given. The chromatic number when is a power of a prime is computed. Further properties for and are also discussed.

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).

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）.

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 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.

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 .

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.

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.

Vertex-distinguishing E-total Coloring of Complete Bipartite Graph K 7,n when7≤n≤95

Let G be a simple graph. A total coloring f of G is called an E-total coloring if no two adjacent vertices of G receive the same color, and no edge of G receives the same color as one of its endpoints. For an E-total coloring f of a graph G and any vertex x of G, let C（x） denote the set of colors of vertex x and of the edges incident with x, we call C（x） the color set of x. If C（u） ≠ C（v） for any two different vertices u and v of V （G）, then we say that f is a vertex-distinguishing E-total coloring of G or a VDET coloring of G for short. The minimum number of colors required for a VDET coloring of G is denoted by Хvt^e（G） and is called the VDE T chromatic number of G. The VDET coloring of complete bipartite graph K7,n （7 ≤ n ≤ 95） is discussed in this paper and the VDET chromatic number of K7,n （7 ≤ n ≤ 95） has been obtained.

On the Maximum Number of Dominating Classes in Graph Coloring

We investigate the dominating-c-color number,, of a graph G. That is the maximum number of color classes that are also dominating when G is colored using colors. We show that where is the join of G and . This result allows us to construct classes of graphs such that and thus provide some information regarding two questions raised in [1] and [2].

ON THE TOTAL COLORING OF GRAPH G ∨H

The total chromatic number XT(G) of graph G is the least number of colorsassigned to VE(G) such that no adjacent or incident elements receive the same color.Gived graphs G1,G2, the join of G1 and G2, denoted by G1∨G2, is a graph G, V(G) =V(GI)∪V(G2) and E(G) = E(G1)∪E(G2) ∪{uv | u∈(G1), v ∈ V(G2)}. In this paper, it's proved that if v(G) = v(H), both Gc and Hc contain perfect matching and one of the followings holds: (i)Δ(G) =Δ(H) and there exist edge e∈ E(G), e' E E(H)such that both G-e and H-e' are of Class l; (ii)Δ(G)<Δ(H) and there exixst an edge e ∈E(H) such that H-e is of Class 1, then, the total coloring conjecture is true for graph G ∨H.

Vertex-distinguishing VE-total Colorings of Cycles and Complete Graphs

Let G be a simple graph of order at least 2.A VE-total-coloring using k colors of a graph G is a mapping f from V (G) E(G) into {1,2,···,k} such that no edge receives the same color as one of its endpoints.Let C(u)={f(u)} {f(uv) | uv ∈ E(G)} be the color-set of u.If C(u)=C(v) for any two vertices u and v of V (G),then f is called a k-vertex-distinguishing VE-total coloring of G or a k-VDVET coloring of G for short.The minimum number of colors required for a VDVET coloring of G is denoted by χ ve vt (G) and it is called the VDVET chromatic number of G.In this paper we get cycle C n,path P n and complete graph K n of their VDVET chromatic numbers and propose a related conjecture.

Vertex-distinguishing IE-total Colorings of Complete Bipartite Graphs K8,n

Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. For each vertex x of G, let C(x) be the set of colors of vertex x and edges incident to x under f. For an IE-total coloring f of G using k colors, if C(u) ≠ C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-coloring of G or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χ_(vt)^(ie) (G) and is called vertex-distinguishing IE-total chromatic number or the VDIET chromatic number of G for short. The VDIET colorings of complete bipartite graphs K_(8,n)are discussed in this paper. Particularly, the VDIET chromatic number of K_(8,n) are obtained.

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.

A SEVEN-COLOR THEOREM ON EDGE-FACE COLORING OF PLANE GRAPHS

Melnikov(1975) conjectured that the edges and faces of a plane graph G can be colored with △(G) + 3 colors so that any two adjacent or incident elements receive distinct colors, where △(G) denotes the maximum degree of G. This paper proves the conjecture for the case △(G) ≤4.