D(β)-vertex-distinguishing total coloring of graphs

A new concept of the D(β)-vertex-distinguishing total coloring of graphs, i.e., the proper total coloring such that any two vertices whose distance is not larger thanβhave different color sets, where the color set of a vertex is the set composed of all colors of the vertex and the edges incident to it, is proposed in this paper. The D(2)-vertex-distinguishing total colorings of some special graphs are discussed, meanwhile, a conjecture and an open problem are presented.     

关于图的星色数的一点注记

A star coloring of an undirected graph G is a proper coloring of G such that no path of length 3 in G is bicolored.The star chromatic number of an undirected graph G,denoted by χs(G),is the smallest integer k for which G admits a star coloring with k colors.In this paper,we show that if G is a graph with maximum degree △,then χs(G) ≤ [7△3/2],which gets better bound than those of Fertin,Raspaud and Reed.     

圈和完全图的点可区别VE-全染色(英文)

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) 6= 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 χievt(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 K8,n are discussed in this paper. Particularly, the VDIET chromatic number of K8,n are obtained.     

不含相邻三角形的平面图的无圈边着色

An acyclic edge coloring of a graph G is a proper edge coloring such that there are no bichromatic cycles.The acyclic edge chromatic number of a graph G is the minimum number k such that there exists an acyclic edge coloring using k colors and is denoted by χ’ a(G).In this paper we prove that χ ’ a(G) ≤(G) + 5 for planar graphs G without adjacent triangles.     

Vertex-distinguishing E-total Coloring of Complete Bipartite Graph K_(7,n) when 7≤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 χ_(vt)~e(G) and is called the VDET chromatic number of G. The VDET coloring of complete bipartite graph K_(7,n)(7 ≤ n ≤ 95) is discussed in this paper and the VDET chromatic number of K_(7,n)(7 ≤ n ≤ 95) has been obtained.     

联图C_mVG_n的邻点可区别Ｅ-全染色(英文)

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.     

关于完全二部图$K_{4, n}$和$K_{n, n}$的点可区别 $IE$-全染色的一些注记

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 χ ie vt (G),and it is called the VDIET chromatic number of G.We will give VDIET chromatic numbers for complete bipartite graph K4,n (n≥4),K n,n (5≤ n ≤ 21) in this article.     

Remarks on Vertex-Distinguishing IE-Total Coloring of Complete Bipartite Graphs K4,n and Kn,n

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 χ ie vt (G),and it is called the VDIET chromatic number of G.We will give VDIET chromatic numbers for complete bipartite graph K4,n (n≥4),K n,n (5≤ n ≤ 21) in this article.     

若干笛卡尔积图的邻点可区别关联染色

An adjacent vertex distinguishing incidence coloring of graph G is an incidence coloring of G such that no pair of adjacent vertices meets the same set of colors.We obtain the adjacent vertex distinguishing incidence chromatic number of the Cartesian product of a path and a path,a path and a wheel,a path and a fan,and a path and a star.     

$m$个$C_{3}$的并和$m$个$C_{4}$的并的点可区别 $E$-全染色

Let G be a simple graph. A total coloring f of G is called 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 E-total-coloring f of a graph G and any vertex u of G, let Cf （u） or C（u） denote the set of colors of vertex u and the edges incident to u. We call C（u） the color set of u. 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 colorings of G is denoted by X^evt（G）, and it is called the VDET chromatic number of G. In this article, we will discuss vertex-distinguishing E-total colorings of the graphs mC3 and mC4.     

Total coloring of graphs embedded in surfaces of nonnegative Euler characteristic

Let G be a graph which can be embedded in a surface of nonnegative Euler characteristic.In this paper,it is proved that the total chromatic number of G is △(G)+1 if △(G)9,where △(G)is the maximum degree of G.     

Planar graphs with maximum degree 8 and without intersecting chordal 4-cycles are 9-totally colorable

The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ’’ (G). It is shown that if a planar graph G has maximum degree Δ≥9, then χ’’ (G) = Δ + 1. In this paper, we prove that if G is a planar graph with maximum degree 8 and without intersecting chordal 4-cycles, then χ ’’(G) = 9.     

乘积图的Fractional控制

Let G =(V,E) be a simple graph.For any real function g :V-→ R and a subset S V,we write g(S) =∑v∈Sg(v).A function f :V-→ [0,1] is said to be a fractional dominating function(F DF) of G if f(N [v]) ≥ 1 holds for every vertex v ∈ V(G).The fractional domination number γf(G) of G is defined as γf(G) = min{f(V)|f is an F DF of G }.The fractional total dominating function f is defined just as the fractional dominating function,the difference being that f(N(v)) ≥ 1 instead of f(N [v]) ≥ 1.The fractional total domination number γ0f(G) of G is analogous.In this note we give the exact values ofγf(Cm × Pn) and γ0f(Cm × Pn) for all integers m ≥ 3 and n ≥ 2.     

一类正则循环图的Cartesian积图的D(β)-点可区别VI-全染色及I-全染色

设G是简单图,若图G的全染色f满足:1)(?)uv,vw∈E(G),有f(uv)≠f(vw);2)(?)uv∈E(G),u≠v,有f(u)≠f(v);3)(?)u,v∈V(G),0相似文献   

关于图的符号边全控制数

Let G = （V,E） be a graph.A function f ： E → {-1,1} is said to be a signed edge total dominating function （SETDF） of G if e ∈N（e） f（e ） ≥ 1 holds for every edge e ∈ E（G）.The signed edge total domination number γ st （G） of G is defined as γ st （G） = min{ e∈E（G） f（e）｜f is an SETDF of G}.In this paper we obtain some new lower bounds of γ st （G）.     

路与圈图的联图的均匀强边染色

For a proper edge coloring c of a graph G,if the sets of colors of adjacent vertices are distinct,the edge coloring c is called an adjacent strong edge coloring of G.Let c i be the number of edges colored by i.If |c i c j | ≤ 1 for any two colors i and j,then c is an equitable edge coloring of G.The coloring c is an equitable adjacent strong edge coloring of G if it is both adjacent strong edge coloring and equitable edge coloring.The least number of colors of such a coloring c is called the equitable adjacent strong chromatic index of G.In this paper,we determine the equitable adjacent strong chromatic index of the joins of paths and cycles.Precisely,we show that the equitable adjacent strong chromatic index of the joins of paths and cycles is equal to the maximum degree plus one or two.     

星色数在三与四之间的一些平面图类

本文构造出了星色数为３＋１／ｄ,３＋2（2ｄ－１）,３＋３／（３ｄ－１）,和３＋３／（３ｄ－２）的一些平面图类,从而部分解决了Ｖｉｎｃｅ的问题.     

一类正则循环图的Cartesian积图的D(β)-点可区别Ⅵ-全染色及Ⅰ-全染色

设G是简单图,若图G的全染色f满足:1)(V)uv,vw∈E(G),有f(uv)≠f(vw);2)(V)uv∈E(G),u≠v,有f(u)≠f(v);3)(V)u,v∈V(G),0＜d(u,v)≤β,有S(u)≠S(v),这里色集合S(u)={f(u)}∪{f(uv) |uv∈E(G)}.则称f是图G的一个D(β)-点可区别Ⅰ-全染色.若f只满足条件1)和3),则称f是图G的一个D(β)-点可区别Ⅵ-全染色.研究了当β=1,2时一类正则循环图与圈的Cartesian积图的D(β)-点可区别Ⅵ-全色数和D(β)-点可区别Ⅰ-全色数,并讨论了正则图的D(β)-点可区别Ⅵ-全色数和D(β)-点可区别Ⅰ-全色数的上界.     

C_m·S_n的D(2)-点可区别边色数

对阶数不小于3的连通图G(V,E),设α,β为正整数,令映射f:Ef{1,2,…,α},若u,v∈V(G),1≤d(u,v)≤β,有C(u)≠C(v),则称f为G的一个α-D(β)-点可区别的边染色,简记为α-D(β)-VDPEC,对一个图进行α-D(β)-点可区别的边染色,所需的最少的颜色数称为图G的D(β)-点可区别的边色数,记为χ′β-vd(G),其中d(u,v)表示两个点u,v之间的最短距离.得到了Cm.Sn的D(2)-点可区别边色数.