Neighbor sum distinguishing total colorings of triangle free planar graphs

A total k-coloring c of a graph G is a proper total coloring c of G using colors of the set[k] = {1, 2,..., k}. Let f(u) denote the sum of the color on a vertex u and colors on all the edges incident to u. A k-neighbor sum distinguishing total coloring of G is a total k-coloring of G such that for each edge uv ∈ E(G), f(u) = f(v). By χ nsd(G), we denote the smallest value k in such a coloring of G. Pil′sniak and Wo′zniak conjectured that χ nsd(G) ≤Δ(G) + 3 for any simple graph with maximum degree Δ(G). In this paper, by using the famous Combinatorial Nullstellensatz, we prove that the conjecture holds for any triangle free planar graph with maximum degree at least 7.     

图K_(3,4)∨K_t的点可区别正常边染色

设f是图G的一个正常边染色.对任意x∈V(G),令S(x)表示与点x相关联的边的颜色所构成的集合.若对任意u,v∈V(G),u≠v,有S(u)≠S(v),则称f是图G的一个点可区别正常边染色.对一个图G进行点可区别正常边染色所需的最少的颜色的数目称为G的点可区别正常边色数,记为χ_s'(G).讨论了图K_(3,4)∨K_t的点可区别正常边染色及其色数,利用正多边形的对称性构造染色以及组合分析的方法,确定了图K_(3,4)∨K_t的点可区别正常边色数,得到了当t是大于等于2的偶数以及t是奇数且3≤t≤25时,χ_s'(K_(3,4)∨K_t)=t+7;当t是奇数且t≥27时,χ_s'(K_(3,4)∨K_t)=t+8.     

关于图的点可区别边染色猜想的一点注

图G的一个k-正常边染色f被称为点可区别的是指任意两点的点及其关联边所染色集合不同,所用最少颜色数被称为G的点可区别边色数,张忠辅教授提出一个猜想即对每一个正整数k≥3,总存在一个最大度为△(G)=k≥3的图G,图G一定有一个子图H,使得G的点可区别的边色数不超过子图的.本文证明了对于最大度△≤6时,猜想正确.     

图的邻点可区别无圈边染色的一个界

图G的一个正常边染色被称作邻点可区别无圈边染色,如果G中无二色圈,且相邻点关联边的色集合不同.应用概率的方法得到了图G的一个邻点可区别无圈边色数的上界,其中图G为无孤立边的图.     

关于图K_(2n)-E(C_4)的点可区别边色数

图G的一个k-正常边染色f被称为点可区别边染色是指任何两点的点及其关联边的色集合不同,所用最小的正整数k被称为G的点可区别边色数,记为x′_(vd)(G).用K_(2n)-E(C_4)表示2n阶完全图删去其中一条4阶路的边后得到的图,文中得到了K_(2n)-E(_4)的点可区别边色数.     

路的字典积的邻和可区别边染色

图G的正常[k]-边染色σ是指颜色集合为[k]={1,2,…,k}的G的一个正常边染色.用w_σ（x）表示顶点x关联边的颜色之和,即w_σ（x）=∑_e∋x σ（e）,并称w_σ（x）关于σ的权.图G的k-邻和可区别边染色是指相邻顶点具有不同权的正常[k]-边染色,最小的k值称为G的邻和可区别边色数,记为χ'_∑（G）.现得到了路P_n与简单连通图H的字典积P_n[H]的邻和可区别边色数的精确值,其中H分别为正则第一类图、路、完全图的补图.     

图K_3~cVK_t的点可区别正常边染色

图G的正常边染色称为是点可区别的,如果对G的任意两顶点的关联边的颜色构成的集合不同.对图G进行点可区别正常边染色所需要的最少颜色数称为图G的点可区别正常边色数,记为x_s'(G).给出了3阶空图与t阶完全图的联图的点可区别正常边色数.     

图C_5∨K_t的邻点可区别全色数

设f是图G的一个正常全染色.对任意x∈V(G),令C(x)表示与点x相关联的边的颜色以及点x的颜色所构成的集合.若对任意uv∈E(G),有C(u)≠C(v),则称.f是图G的一个邻点可区别全染色.对一个图G进行邻点可区别全染色所需的最少的颜色的数目称为G的邻点可区别全色数,记为Xat(G).用C_5∨K_t表示长为5的圈与t阶完全图的联图.讨论了C_5∨K_t的邻点可区别全色数.利用正多边形的对称性构造染色以及组合分析的方法,得到了当t是大于等于3的奇数以及t是偶数且2≤t≤22时,X_(at)(C_5 V K_t)=t+6,当t是偶数且t≥24时,X_(at)(C_5 V K_t)=t+7.     

某些中间图的邻点可区别E-全色数

设G（V,E）是简单连通图,T（G）为图G的所有顶点和边构成的集合,并设C是k-色集（k是正整数）,若T（G）到C的映射f满足：对任意uv∈E（G）,有f（u）≠f（v）,f（u）≠f（uv）,f（v）≠f（uv）,并且C（u）≠C（v）,其中C（u）={f（u）}∪{f（uv）｜uv∈E（G）}.那么称f为图G的邻点可区别E-全染色（简记为k-AVDETC）,并称χ_（at）~e（G）=min{k｜图G有k-邻点可区别E-全染色}为G的邻点可区别E-全色数.图G的中间图M（G）就是在G的每一个边上插入一个新的顶点,再把G上相邻边上的新的顶点相联得到的.探讨了路、圈、扇、星及轮的中间图的邻点可区别E-全染色,并给出了这些中间图的邻点可区别E-全色数.     

Halin-图的邻强边染色

图G(V,E)的正常κ-边染色f叫做图G(V,E)的κ-邻强边染色当且仅当任意uv∈E(G)满足f[u]≠f[v],其中,f[u]={f(uw)|uw∈E(G)},称f是G的κ-临强边染色,简记为κ-ASEC.并且x′_as(G)=min{k|κ-ASEC of G}叫做G(V,E)的邻强边色数.本文研究了△(G)≥5的Halin-图的邻强边色数.     

关于积图点可区别边染色的若干结论

如果图G的一个正常边染色满足任意两个不同点的关联边色集不同, 则称为点可区别边染色(VDEC), 其所用最少颜色数称为点可区别边色数. 利用构造法给出了积图点可区别边染色的一个结论, 得到了关于积图点可区别边色数的若干结果, 并且给出25个具体积图的点可区别边色数, 验证了它们满足点可区别边染色猜想(VDECC).     

Neighbor sum distinguishing edge coloring of subcubic graphs

A proper edge-k-coloring of a graph G is a mapping from E(G) to {1, 2,..., k} such that no two adjacent edges receive the same color. A proper edge-k-coloring of G is called neighbor sum distinguishing if for each edge uv ∈ E(G), the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. Let χ_Σ'(G) denote the smallest value k in such a coloring of G. This parameter makes sense for graphs containing no isolated edges(we call such graphs normal). The maximum average degree mad(G) of G is the maximum of the average degrees of its non-empty subgraphs. In this paper, we prove that if G is a normal subcubic graph with mad(G) 5/2,then χ_Σ'(G) ≤ 5. We also prove that if G is a normal subcubic graph with at least two 2-vertices, 6 colors are enough for a neighbor sum distinguishing edge coloring of G, which holds for the list version as well.     

图 P2×Cn的均匀邻强边色数

对图G(V,E),一正常边染色f若满足(1)对(V)uv∈E(G),f[u]≠f[v],其中f[u]={f(uv)|uv∈E};(2)对任意i≠j,有||E|-|Ej||≤1,其中Ei={e| e∈E(G)且f(e)=i}.则称f为G(V,E)的一k-均匀邻强边染色,简称k-EASC,并且称Xcas(G)=min{k|存在G(V,E)的一k-EASC为G(V,E)的均匀邻强边色数.本文得到了图P2×Cn的均匀邻强边色数.     

完全图及完全二部图的点可区别Ⅳ-全染色

设G是简单图,图G的一个k-点可区别Ⅵ-全染色(简记为k-VDIVT染色),f是指一个从V(G)∪E(G)到{1,2,…,k}的映射,满足:()uv,uw∈E(G),v≠w,有,f(uv)≠f(uw);()u,V∈V(G),u≠v,有C(u)≠C(v),其中C(u)={f(u)}∪{f(uv)|uv∈E(G)}.数min{k|G有一个k-VDIVT染色}称为图G的点可区别Ⅵ-全色数,记为x_(vt)~(iv)(G).讨论了完全图K_n及完全二部图K_(m,n)的VDIVT色数.     

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.     

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.     

图K_(2n)\E(F_4)(n≥12)的点可区别边染色

对简单图G(V,E),设f是从E(G)到{1,2,…,k}的映射,k为自然数,如果.f满足:1)对任意的uv,uw∈E(G),v≠w,有.f(uv)≠f(uw);2)对任意的u,v∈V(G),u≠v,有C(u)≠C(v).则称f为图G的k-点可区别边染色法,而最小的k被称为点可区别边色数(其中C(u)={f(uv)|uv∈E(G)}.研究了图K_(2n)\E(F_4)(n≥12)的点可区别边色数.     

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.     

△(G)=4的Halin-图的邻强边染色

图G(V,E)的一正常k-边染色f称为G(V,E)的一k-邻强边染色(简称k-ASEC)当且仅当任意uv∈E(G)满足f[u]≠f[v],其中f[u]={f(uw)|uw∈E(G)},并称Xas(G)=min{k|存在G的一k-ASEC}为G的邻强边色数.本文研究了△(G)=4的Halin-图的邻强边染色,得到了如下结果对△(G)=4的Halin-图有△(G)=4≤Xas(G)≤△(G)+1=5.     

Total coloring of planar graphs without chordal 7-cycles

A k-total-coloring of a graph G is a coloring of vertices and edges of G using k colors such that no two adjacent or incident elements receive the same color.In this paper,it is proved that if G is a planar graph with Δ(G) ≥ 7 and without chordal 7-cycles,then G has a(Δ(G) + 1)-total-coloring.