On the adjacent vertex-distinguishing acyclic edge coloring of some graphs

A proper edge coloring of a graph G is called adjacent vertex-distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the coloring set of edges incident with u is not equal to the coloring set of edges incident with v, where uv∈ E（G）. The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by X＇Aa（G）, is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. If a graph G has an adjacent vertex distinguishing acyclic edge coloring, then G is called adjacent vertex distinguishing acyclic. In this paper, we obtain adjacent vertex-distinguishing acyclic edge coloring of some graphs and put forward some conjectures.

Smarandachely Adjacent-vertex-distinguishing Proper Edge Coloring ofK4 V Kn

Let f be a proper edge coloring of G using k colors. For each x ∈ V（G）, the set of the colors appearing on the edges incident with x is denoted by Sf（x） or simply S（x） if no confusion arise. If S（u） = S（v） and S（v） S（u） for any two adjacent vertices u and v, then f is called a Smarandachely adjacent vertex distinguishing proper edge col- oring using k colors, or k-SA-edge coloring. The minimum number k for which G has a Smarandachely adjacent-vertex-distinguishing proper edge coloring using k colors is called the Smarandachely adjacent-vertex-distinguishing proper edge chromatic number, or SA- edge chromatic number for short, and denoted by Xsa（G）. In this paper, we have discussed the SA-edge chromatic number of K4 V Kn.

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.

Vertex-distinguishing Total Colorings of 2Cn

Let f be a proper total k-coloring of a simple graph G. For any vertex x ∈ V（G）, let Cf（x） denote the set of colors assigned to vertex x and the edges incident with x. If Cf（u） ≠ Cf（v） for all distinct vertices u and v of V（G）, then f is called a vertex- distinguishing total k-coloring of G. The minimum number k for which there exists a vertex- distinguishing total k-coloring of G is called the vertex-distinguishing total chromatic number of G and denoted by Xvt（G）. The vertex-disjoint union of two cycles of length n is denoted by 2Cn. We will obtain Xvt（2Cn） in this paper.

Vertex-distinguishing IE-total Colorings of Cycles and Wheels

Let G be a simple graph. An IE-total coloring f of G refers to a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. Let C（u） be the set of colors of vertex u and edges incident to u 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 V （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 χievt（G）, and is called the VDIET chromatic number of G. We get the VDIET chromatic numbers of cycles and wheels, and propose related conjectures in this paper.

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.

Adjacent vertex-distinguishing total colorings of K_s∨K_t

Let G be a simple graph and f be a proper total kcoloring of G. The color set of each vertex v of G is the set of colors appearing on v and the edges incident to v. The coloring f is said to be an adjacent vertex-distinguishing total coloring if the color sets of any two adjacent vertices are distinct. The minimum k for which such a coloring of G exists is called the adjacent vertex-distinguishing total chromatic number of G. The join graph of two vertex-disjoint graphs is the graph union of these two graphs together with all the edges that connect the vertices of one graph with the vertices of the other. The adjacent vertex-distinguishing total chromatic numbers of the join graphs of an empty graph of order s and a complete graph of order t are determined.

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.

Adjacent Vertex-distinguishing E-total Coloring on Some Join Graphs Cm V Gn

Let G(V, E) be a simple connected graph and k be positive integers. A mapping f from V∪E to {1, 2, ··· , k} is called an adjacent vertex-distinguishing E-total coloring of G(abbreviated to k-AVDETC), if for uv ∈ E(G), we have f(u) ≠ f(v), f(u) ≠ f(uv), f(v) ≠ f(uv), C(u) ≠C(v), where C(u) = {f(u)}∪{f(uv)|uv ∈ E(G)}. The least number of k colors required for which G admits a k-coloring is called the adjacent vertex-distinguishing E-total chromatic number of G is denoted by x^e_(at) (G). In this paper, the adjacent vertexdistinguishing E-total colorings of some join graphs C_m∨G_n are obtained, where G_n is one of a star S_n , a fan F_n , a wheel W_n and a complete graph K_n . As a consequence, the adjacent vertex-distinguishing E-total chromatic numbers of C_m∨G_n are confirmed.

ON EQUITABLE VERTEX DISTINGUISHING EDGE COLORINGS OF TREES

It has been known that determining the exact value of vertex distinguishing edge index X ＇8（G） of a graph G is difficult, even for simple classes of graphs such as paths, cycles, bipartite complete graphs, complete, graphs, and graphs with maximum degree 2. Let rid（G） denote the number of vertices of degree d in G, and let X＇es（G） be the equitable vertex distinguishing edge index of G. We show that a tree T holds nl （T） ≤ X ＇s （T） ≤ n1 （T） ＋ 1 and X＇s（T） = X＇es（T） if T satisfies one of the following conditions （i） n2（T） ≤△（T） or （ii） there exists a constant c with respect to 0 〈 c 〈 1 such that n2（T） △ cn1（T） and ∑3 ≤d≤△（T）nd（T） ≤ （1 - c）n1（T） ＋ 1.

Algorithm on the Optimal Vertex-Distinguishing Total Coloring of mC9

Let G be a simple graph and f be a proper total coloring(or a total coloring in brief) of G. For any vertex u in G, Cf(u) denote the set of colors of vertex u and edges which incident with vertex u. Cf(u) is said to be the color set of vertex u under f. If Cf(u) = Cf(v)for any two distinct vertices u and v of G, then f is called vertex-distinguishing total coloring of G(in brief VDTC), a vertex distinguishing total coloring using k colors is called k-vertexdistinguishing total coloring of G(in brief k-VDTC). The minimum number k for which there exists a k-vertex-distinguishing total coloring of G is called the vertex-distinguishing total chromatic number of G, denoted by χvt(G). By the method of prior distributing the color sets, we obtain vertex-distinguishing total chromatic number of m C9 in this paper.

Some Results on Edge Cover Coloring of Double Graphs

Let G be a simple graph with vertex set V(G) and edge set E(G). An edge coloring C of G is called an edge cover coloring, if each color appears at least once at each vertex . The maximum positive integer k such that G has a k edge cover coloring is called the edge cover chromatic number of G and is denoted by . It is known that for any graph G, . If , then G is called a graph of CI class, otherwise G is called a graph of CII class. It is easy to prove that the problem of deciding whether a given graph is of CI class or CII class is NP-complete. In this paper, we consider the classification on double graph of some graphs and a polynomial time algorithm can be obtained for actually finding such a classification by our proof.

The 4-Acyclic Edge Coloring of Graphs with Large Girths

A proper edge coloring of a graph is acyclic, if every cycle of the graph has at least 3 colors. Let r be a positive integer. An edge coloring is r-acyclic if it is proper and every cycle C has at least ?colors. The r-acyclic edge chromatic ??number of a graph G?is the minimum number of colors needed for any r-acyclic edge coloring of G. When r=4, the result of this paper is that the 4-acyclic chromatic number of a graph with maximum degree Δ and girth ?is less than 18Δ. Furthermore, if the girth of graph G?is at least , then .

A Non-Conventional Coloring of the Edges of a Graph

Coloring the nodes of a graph is a commonly used technique to speed up clique search algorithms. Coloring the edges of the graph as a preconditioning method can also be used to speed up computations. In this paper we will show that an unconventional coloring scheme of the edges leads to an NP-complete problem when one intends to determine the optimal number of colors.

On a Number of Colors in Cyclically Interval Edge Colorings of Simple Cycles

A proper edge t-coloring of a graph G is a coloring of its edges with colors 1,2,???,t such that all colors are used, and no two adjacent edges receive the same color. A cyclically interval t-coloring of a graph G is a proper edge t-coloring of G such that for each its vertex x, either the set of colors used on edges incident to x or the set of colors not used on edges incident to x forms an interval of integers. For an arbitrary simple cycle, all possible values of t are found, for which the graph has a cyclically interval t-coloring.

Edge Colorings of Planar Graphs without 6-Cycles with Two Chords

It is proved here that if a planar graph has maximum degree at least 6 and any 6-cycle contains at most one chord, then it is of class 1.

Edge Coloring of Graphs with Applications in Coding Theory

In this paper,a new type of edge coloring of graphs together with an algorithm for such an edge coloring is presented to construct some columnweight three low-density parity-check(LDPC)codes whose Tanner graphs are free of 4-cycles.This kind of edge coloring is applied on some well-known classes of graphs such as complete graphs and complete bipartite graphs to generate some column-weight 3 LDPC codes having flexibility in terms of code length and rate.Interestingly,the constructed(3;k)-regular codes with regularities k=4;5;:::;22 have lengths n=12;20;26,35;48;57;70;88;104;117;140;155;176;204;228;247;280;301;330;having minimum block length compared to the best known similar codes in the literature.In addition to linear complexity of generating such parity-check matrices,they can be considered as the base matrices of some quasi-cyclic(QC)LDPC codes with maximum achievable girth 18,which inherit the low-complexity encoder implementations of QC-LDPC codes.Simulation results show that the QC-LDPC codes with large girth lifted from the constructed base matrices have good performances and outperform random codes,progressive edge growth LDPC codes,some finite fields and group rings based QC-LDPC codes and also have a close competition to the standard IEEE 802.16e(WiMAX)code.

Note on Cyclically Interval Edge Colorings of Simple Cycles

A proper edge t-coloring of a graph G is a coloring of its edges with colors 1, 2,..., t, such that all colors are used, and no two adjacent edges receive the same color. A cyclically interval t-coloring of a graph G is a proper edge t-coloring of G such that for each vertex, either the set of colors used on edges incident to x or the set of colors not used on edges incident to x forms an interval of integers. In this paper, we provide a new proof of the result on the colors in cyclically interval edge colorings of simple cycles which was first proved by Rafayel R. Kamalian in the paper “On a Number of Colors in Cyclically Interval Edge Colorings of Simple Cycles, Open Journal of Discrete Mathematics, 2013, 43-48”.

Note on 2-edge-colorings of complete graphs with small monochromatic k-connected subgraphs

Bollobas and Gyarfas conjectured that for n 〉 4（k - 1） every 2-edge-coloring of Kn contains a monochromatic k-connected subgraph with at least n - 2k ＋ 2 vertices. Liu, et al. proved that the conjecture holds when n 〉 13k - 15. In this note, we characterize all the 2-edge-colorings of Kn where each monochromatic k-connected subgraph has at most n - 2k ＋ 2 vertices for n ≥ 13k - 15.

Cost Edge-Coloring of a Cactus

Let C be a set of colors, and let ?be an integer cost assigned to a color c in C. An edge-coloring of a graph ?is assigning a color in C to each edge ?so that any two edges having end-vertex in common have different colors. The cost ?of an edge-coloring f of G is the sum of costs ?of colors ?assigned to all edges e in G. An edge-coloring f of G is optimal if ?is minimum among all edge-colorings of G. A cactus is a connected graph in which every block is either an edge or a cycle. In this paper, we give an algorithm to find an optimal edge- ??coloring of a cactus in polynomial time. In our best knowledge, this is the first polynomial-time algorithm to find an optimal edge-coloring of a cactus.