Zero-divisor Labelings of Graphs

This paper introduces the notions of a zero-divisor labeling and the zero-divisor index of a graph using the zero-divisors of a commutative ring. Viewed in this way, the usual zero-divisor graph is a maximal graph with respect to a zero-divisor labeling. We also study optimal zero-divisor labelings of a finite graph.     

Antimagic Labelings of Join Graphs

An antimagic labeling of a graph with q edges is a bijection from the set of edges of the graph to the set of positive integers \({\{1, 2,\dots,q\}}\) such that all vertex weights are pairwise distinct, where a vertex weight is the sum of labels of all edges incident with the vertex. The join graph G + H of the graphs G and H is the graph with \({V(G + H) = V(G) \cup V(H)}\) and \({E(G + H) = E(G) \cup E(H) \cup \{uv : u \in V(G) {\rm and} v \in V(H)\}}\). The complete bipartite graph K _m,n is an example of join graphs and we give an antimagic labeling for \({K_{m,n}, n \geq 2m + 1}\). In this paper we also provide constructions of antimagic labelings of some complete multipartite graphs.     

On Partitional Labelings of Graphs

The notion of partitional graphs, a subclass of sequential graphs, is introduced, and the cartesian product of a partitional graph and K ₂ is shown to be partitional. Every sequential graph is harmonious and felicitous. The partitional property of some bipartite graphs including the n-dimensional cube Q _n is studied, and thus this paper extends what was known about the sequentialness, harmoniousness and felicitousness of such graphs.     

Graceful Labelings of Nearly Complete Graphs

It is well known that the complete graph K_n is graceful only if n ≤ 4. We prove that K_n — e is graceful only if n ≤ 5 and that any K_n—2e and any K_n — 3e is graceful only if n ≤ 6. Moreover we determine all graceful graphs K_n — G, if G is a star K_1,a with a ≤ n - 2, and if G is a matching M_a with 2a ≤ n. Furthermore we give graceful labelings for the complete multipartite graphs K_1,a,b, K_2,a,b K_1,1,a,b and we conjecture that these together with K_a,b are the only complete multipartite graphs which are graceful.     

Neighbor sum distinguishing total colorings of K 4-minor free graphs

A total [k]-coloring of a graph G is a mapping φ： V（G） U E（G） →{1, 2, ..., k} such that any two adjacent elements in V（G）UE（G） receive different colors. Let f（v） denote the sum of the colors of a vertex v and the colors of all incident edges of v. A total [k]-neighbor sum distinguishing-coloring of G is a total [k]-coloring of G such that for each edge uv E E（G）, f（u） ≠ f（v）. By tt [G, Xsd（ J, we denote the smallest value k in such a coloring of G. Pilniak and Woniak conjectured X＇sd（G） 〈 A（G） ＋ 3 for any simple graph with maximum degree A（G）. This conjecture has been proved for complete graphs, cycles, bipartite graphs, and subcubic graphs. In this paper, we prove that it also holds for Ka-minor free graphs. Furthermore, we show that if G is a Ka-minor flee graph with A（G） 〉 4, then ＂ Xnsd（G） 〈 A（G） ＋ 2. The bound A（G） ＋ 2 is sharp.     

On the Mean Distance in Scale Free Graphs

We consider a graph, where the nodes have a pre-described degree distribution F, and where nodes are randomly connected in accordance to their degree. Based on a recent result (R. van der Hofstad, G. Hooghiemstra and P. Van Mieghem, “Random graphs with finite variance degrees,” Random Structures and Algorithms, vol. 17(5) pp. 76–105, 2005), we improve the approximation of the mean distance between two randomly chosen nodes given by M. E. J. Newman, S. H. Strogatz, and D. J. Watts, “Random graphs with arbitrary degree distribution and their application,” Physical Review. E vol. 64, 026118, pp. 1–17, 2001. Our new expression for the mean distance involves the expectation of the logarithm of the limit of a super-critical branching process. We compare simulations of the mean distance with the results of Newman et al. and with our new approach. AMS 2000 Subject Classification: 05C80, 60F05     

Vertex Partitions of K4,4-Minor Free Graphs

 We prove that a 4-connected K _4,4-minor free graph on n vertices has at most 4n−8 edges and we use this result to show that every K _4,4-minor free graph has vertex-arboricity at most 4. This improves the case (n,m)=(7,3) of the following conjecture of Woodall: the vertex set of a graph without a K _n -minor and without a -minor can be partitioned in n−m+1 subgraphs without a K _m -minor and without a -minor. Received: January 7, 1998 Final version received: May 17, 1999     

On the Edit Distance from ‐Free Graphs

The edit distance between two graphs on the same vertex set is defined to be the size of the symmetric difference of their edge sets. The edit distance function of a hereditary property, , is a function of p, and measures, asymptotically, the furthest graph of edge density p from under this metric. In this article, we address the hereditary property , the property of having no induced copy of the complete bipartite graph with two vertices in one class and t in the other. Employing an assortment of techniques and colored regularity graph constructions, we are able to determine the edit distance function over the entire domain when and extend the interval over which the edit distance function for is known for all values of t, determining its maximum value for all odd t. We also prove that the function for odd t has a nontrivial interval on which it achieves its maximum. These are the only known principal hereditary properties for which this occurs. In the process of studying this class of functions, we encounter some surprising connections to extremal graph theory problems, such as strongly regular graphs and the problem of Zarankiewicz.     

Super-edge-graceful Labelings of Some Cubic Graphs

The notion of super-edge-graceful graphs was introduced by Mitchem and Simoson in 1994.However, few examples except trees are known. In this paper, we exhibit two classes of infinitely many cubic graphs which are super-edge-graceful. A conjecture is proposed.     

Distance Magic Labeling in Complete 4-partite Graphs

Let G be a complete k-partite simple undirected graph with parts of sizes \(p_1\le p_2\cdots \le p_k\). Let \(P_j=\sum _{i=1}^jp_i\) for \(j=1,\ldots ,k\). It is conjectured that G has distance magic labeling if and only if \(\sum _{i=1}^{P_j} (n-i+1)\ge j{{n+1}\atopwithdelims (){2}}/k\) for all \(j=1,\ldots ,k\). The conjecture is proved for \(k=4\), extending earlier results for \(k=2,3\).     

On embeddings of graphs containing no K5-minor

In this paper, we proved that every 2-connected graph containing no K₅-minor has a closed 2-cell embedding on some 2-manifold surface. © 1996 John Wiley & Sons, Inc.     

Packing Triangles in K 4-Free Graphs

Paul Erd?s conjectured that every K ₄-free graph of order n and size ${k + \lfloor n^2/4\rfloor}$ contains at least k edge disjoint triangles. In this note, we prove that such a graph contains at least 32k/35 + o(n ²) edge disjoint triangles and prove his conjecture for k ≥  0.077n ².     

没有K4-图子式的图的无圈边色数

一个图G 的无圈k- 边染色是指G 的一个正常的不产生双色圈的k- 边染色. G 的无圈边色数a′(G) 定义为使得G 有一个无圈k- 边染色的最小的整数k. 本文完全刻画了最大度不为4 的没有K₄-图子式的图的无圈边色数.     

完全图的Cantesian积的L(2,1)-圆标定

For positive integers j and k with j ≥ k, an L（j, k）-labeling of a graph G is an assignment of nonnegative integers to V（G） such that the difference between labels of adjacent vertices is at least j, and the difference between labels of vertices that are distance two apart is at least k. The span of an L（j, k）-labeling of a graph G is the difference between the maximum and minimum integers it uses. The λj, k-number of G is the minimum span taken over all L（j, k）-labelings of G. An m-（j, k）-circular labeling of a graph G is a function f ： V（G） →{0, 1, 2,..., m - 1} such that ｜f（u） - f（v）｜m ≥ j if u and v are adjacent; and ｜f（u） - f（v）｜m 〉 k ifu and v are at distance two, where ｜x｜m = min{｜xl｜, m-｜x｜}. The minimum integer m such that there exists an m-（j, k）-circular labeling of G is called the σj,k-number of G and is denoted by σj,k（G）. This paper determines the σ2,1-number of the Cartesian product of any three complete graphs.     

两类图的(d,1)-全标号

主要讨论了W_n与C_m的笛卡尔积和均衡完全r-部图K_r(n)的(d,1)-全标号,并得出了(d,1)-全数λ_d~T(W_n□C_m)和λ_d~T(K_(r(n)))的确切值.