Rainbow Numbers for Cycles with Pendant Edges

A subgraph of an edge-colored graph is called rainbow if all of its edges have different colors. For a graph H and a positive integer n, the anti-Ramsey number f (n, H) is the maximum number of colors in an edge-coloring of K _n with no rainbow copy of H. The rainbow number rb(n, H) is the minimum number of colors such that any edge-coloring of K _n with rb(n, H) number of colors contains a rainbow copy of H. Certainly rb(n, H) = f(n, H) + 1. Anti-Ramsey numbers were introduced by Erdős et al. [4] and studied in numerous papers. We show that for n ≥ k + 1, where C _k ⁺ denotes a cycle C _k with a pendant edge.     

Colored Saturation Parameters for Rainbow Subgraphs

Inspired by a 1987 result of Hanson and Toft [Edge‐colored saturated graphs, J Graph Theory 11 (1987), 191–196] and several recent results, we consider the following saturation problem for edge‐colored graphs. An edge‐coloring of a graph F is rainbow if every edge of F receives a different color. Let denote the set of rainbow‐colored copies of F. A t‐edge‐colored graph G is ‐saturated if G does not contain a rainbow copy of F but for any edge and any color , the addition of e to G in color i creates a rainbow copy of F. Let denote the minimum number of edges in an ‐saturated graph of order n. We call this the rainbow saturation number of F. In this article, we prove several results about rainbow saturation numbers of graphs. In stark contrast with the related problem for monochromatic subgraphs, wherein the saturation is always linear in n, we prove that rainbow saturation numbers have a variety of different orders of growth. For instance, the rainbow saturation number of the complete graph lies between and , the rainbow saturation number of an n‐vertex star is quadratic in n, and the rainbow saturation number of any tree that is not a star is at most linear.     

Rainbow faces in edge‐colored plane graphs

A face of an edge‐colored plane graph is called rainbow if the number of colors used on its edges is equal to its size. The maximum number of colors used in an edge coloring of a connected plane graph Gwith no rainbow face is called the edge‐rainbowness of G. In this paper we prove that the edge‐rainbowness of Gequals the maximum number of edges of a connected bridge face factor H of G, where a bridge face factor H of a plane graph Gis a spanning subgraph H of Gin which every face is incident with a bridge and the interior of any one face f∈F(G) is a subset of the interior of some face f′∈F(H). We also show upper and lower bounds on the edge‐rainbowness of graphs based on edge connectivity, girth of the dual graphs, and other basic graph invariants. Moreover, we present infinite classes of graphs where these equalities are attained. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 84–99, 2009     

On the Strongest Form of a Theorem of Whitney for Hamiltonian Cycles in Plane Triangulations

In this article, we investigate hamiltonian cycles in plane triangulations. The aim of the article is to find the strongest possible form of Whitney's theorem about hamiltonian triangulations in terms of the decomposition tree defined by separating triangles. We will decide on the existence of nonhamiltonian triangulations with given decomposition trees for all trees except trees with exactly one vertex with degree and all other degrees at most 3. For these cases, we show that it is sufficient to decide on the existence of nonhamiltonian triangulations with decomposition tree K_{1, 4} or K_{1, 5}. We also give computational results on the size of a possible minimal nonhamiltonian triangulation with these decomposition trees.     

Improved Upper Bounds for Gallai–Ramsey Numbers of Paths and Cycles

Given a graph G and a positive integer k, define the Gallai–Ramsey number to be the minimum number of vertices n such that any k‐edge coloring of contains either a rainbow (all different colored) triangle or a monochromatic copy of G. In this work, we improve upon known upper bounds on the Gallai–Ramsey numbers for paths and cycles. All these upper bounds now have the best possible order of magnitude as functions of k.     

New Lower Bounds on the Multicolor Ramsey Numbers R r (C 2 m )

The multicolor Ramsey number R_r(H) is defined to be the smallest integer n=n(r) with the property that any r-coloring of the edges of the complete graph K_n must result in a monochromatic subgraph of K_n isomorphic to H. It is well known that 2^rm<R_r(C_2m+1)<2(r+2)!m and R_r(C_2m)≥(r−1)(m−1)+1. In this paper, we prove that R_r(C_2m)≥2(r−1)(m−1)+2. This research is supported by NSFC(60373096, 60573022) and SRFDP(20030141003)     

A Note on the Domination Number of Triangulations

In this note, we prove that every triangulation G on any closed surface has domination number at most . This unifies some results on the domination number of a triangulation on a closed surface.     

The Rainbow Vertex-disconnection in Graphs

Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set of G is called rainbow if any two vertices in X have distinct colors. The graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S of G such that when x and y are nonadjacent, S is rainbow and x and y belong to different components of G-S; whereas when x and y are adjacent, S + x or S + y is rainbow and x and y belong to different components of(G-xy)-S. For a connected graph G, the rainbow vertex-disconnection number of G, denoted by rvd(G), is the minimum number of colors that are needed to make G rainbow vertexdisconnected. In this paper, we characterize all graphs of order n with rainbow vertex-disconnection number k for k ∈ {1, 2, n}, and determine the rainbow vertex-disconnection numbers of some special graphs. Moreover, we study the extremal problems on the number of edges of a connected graph G with order n and rvd(G) = k for given integers k and n with 1 ≤ k ≤ n.     

Fan-Type Theorem for Long Cycles Containing a Specified Edge

In this paper, we prove that if G is 3-connected noncomplete graph of order n satisfying min{max{d(u),d(v)}:d(u,v)=2}=μ, then for each edge e, G has a cycle containing e of length at least min{n,2μ}, unless G is a spanning subgraph of K_μ + K^c_n−μ or K₃+(lK_μ−2∪K_s), where n=l(μ−2)+s+3,1≤s≤μ−2. Partially supported by NNSFC(No. 60172005); Partially supported by NNSFC(No. 10431020);     

圈的上定位控制数

A set D of vertices in a graph G = （V, E） is a locating-dominating set （LDS） if for every two vertices u, v of V ／ D the sets N（u） ∩D and N（v） ∩ D are non-empty and different. The locating-domination number γL（G） is the minimum cardinality of an LDS of G, and the upper-locating domination number FL（G） is the maximum cardinality of a minimal LDS of G. In the present paper, methods for determining the exact values of the upper locating-domination numbers of cycles are provided.     

Anti‐Ramsey Problems for t Edge‐Disjoint Rainbow Spanning Subgraphs: Cycles,Matchings, or Trees

We seek the maximum number of colors in an edge‐coloring of the complete graph not having t edge‐disjoint rainbow spanning subgraphs of specified types. Let , , and denote the answers when the spanning subgraphs are cycles, matchings, or trees, respectively. We prove for and for . We prove for and for . We also provide constructions for the more general problem in which colorings are restricted so that colors do not appear on more than q edges at a vertex.     

Triangulations in CGAL

This paper presents the main algorithmic and design choices that have been made to implement triangulations in the computational geometry algorithms library .     

Anti‐Ramsey numbers of doubly edge‐critical graphs

Given a graph H and a positive integer n, Anti‐Ramsey number AR(n, H) is the maximum number of colors in an edge‐coloring of K_n that contains no polychromatic copy of H. The anti‐Ramsey numbers were introduced in the 1970s by Erd?s, Simonovits, and Sós, who among other things, determined this function for cliques. In general, few exact values of AR(n, H) are known. Let us call a graph H doubly edge‐critical if χ(H?e)≥p+ 1 for each edge e∈E(H) and there exist two edges e₁, e₂ of H for which χ(H?e₁?e₂)=p. Here, we obtain the exact value of AR(n, H) for any doubly edge‐critical H when n?n₀(H) is sufficiently large. A main ingredient of our proof is the stability theorem of Erd?s and Simonovits for the Turán problem. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 210–218, 2009     

最大度不小于5的外平面图的邻强边染色

图G(V,E)的一k-正常边染色叫做k-邻强边染色当且仅当对任意uv∈E(G)有,f[u]≠f[v],其中f[u]={f(uw)|uw∈E(G)},f(uw)表示边uw的染色.并且x'as(G)=min{k|存在k-图G的邻强边染色}叫做图G的图的邻强边色数.本文证明了对最大度不小于5的外平面图有△≤x'as(G)≤△ 1,且x'as(G)=△ 1当且仅当存在相邻的最大度点.     

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.     

On vertex-coloring 13-edge-weighting

L. Addario-Berry et al. [Discrete Appl. Math., 2008, 156: 1168-1174] have shown that there exists a 16-edge-weighting such that the induced vertex coloring is proper. In this note, we improve their result and prove that there exists a 13-edge-weighting of a graph G, such that its induced vertex coloring of G is proper. This result is one step close to the original conjecture posed by M. Karoński et al.      

t-Cores for ( Δ + t )-edge-coloring

We extend the edge-coloring notion of core (subgraph induced by the vertices of maximum degree) to $t$-core (subgraph induced by the vertices $v$ with ), and find a sufficient condition for $\left(\mathrm{\Delta }+t\right)$-edge-coloring. In particular, we show that for any $t\ge 0$, if the $t$-core of $G$ has multiplicity at most $t+1$, with its edges of multiplicity $t+1$ inducing a multiforest, then $\chi \prime \text{◂≤▸}\left(G\right)\le \mathrm{\Delta }+t$. This extends previous work of Ore, Fournier, and Berge and Fournier. A stronger version of our result (which replaces the multiforest condition with a vertex-ordering condition) generalizes a theorem of Hoffman and Rodger about cores of $\mathrm{\Delta }$-edge-colorable simple graphs. In fact, our bounds hold not only for chromatic index, but for the fan number of a graph, a parameter introduced by Scheide and Stiebitz as an upper bound on chromatic index. We are able to give an exact characterization of the graphs $H$ such that $\text{◂...▸}\text{Fan}\left(G\right)\le \text{◂+▸}\mathrm{\Delta }\left(G\right)+t$ whenever $G$ has $H$ as its $t$-core.     

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.