Edge Colorings of Embedded Graphs

 In his paper, we discuss under what conditions graphs embedded in a surface Σ will be class one. Received: August 14, 1997 Revised: April 15, 1998     

嵌入曲面的特殊图的边染色

设图G是嵌入到欧拉示性数χ(∑)≥0的曲面上的图,χ′(G)和△(G)分别表示图G的边色数和最大度.将证明如果G满足以下条件:1)△(G)≥5;2)图中3-圈和4-圈不相邻;3)图G中没有5-圈的一次剖分,则有χ′(G)=△(G).     

Edge Coloring of Embedded Graphs with Large Girth

Let G be a simple graph embedded in the surface of Euler characteristic ()0. Denote _e(G), and g the edge chromatic number, the maximum degree and the girth of the graph G, respectively. The paper shows that _e(G)= if 5 and g4, or 4 and g5, or 3 and g9. In addition, if ()>0, then _e(G)= if 3 and g8. Acknowledgments.The authors would like to thank Dr. C.Q. Zhang for carefully reading several versions of this paper during its preparation and for suggesting several stylistic changes that have improved the overall presentation.     

On the Total Coloring of Graphs Embeddable in Surfaces

The paper shows that any graph G with the maximum degree (G) 8, which is embeddable in a surface of Euler characteristic() 0, is totally ((G)+2)-colorable. In general, it is shownthat any graph G which is embeddable in a surface and satisfiesthe maximum degree (G) (20/9) (3–())+1 is totally ((G)+2)-colorable.     

关于图符号的边控制 (英)

设γ's(G)和γ'ι(G)分别表示图G的符号边和局部符号边控制数,本文主要证明了:对任何n阶图G(n≥4),均有γ's(G)≤[11/6n-1]和γ'ι(G)≤2n-4成立,并提出了若干问题和猜想．     

关于图的一般邻点可区别边染色

给出了轮图W_n、扇图F_n、风车图K_2~t、图D_(m,4)、图D_(m,n)、齿轮图W_n的一般邻点可区别色指标.     

On Edge Semi-Isomorphisms and Semi-Dualities of Graphs

Let G₁ and G₂ be undirected graphs, and ?₁(G ₁) and ?₂(G ₂) be families of edge sets of G₁ and G₂, respectively. An (?₁,?₂)-semi-isomorphism ofG ₁ ontoG ₂ is an edge bijection between G₁ and G₂ that induces an injection from ?₁(G ₁) to ?₂(G ₂). This concept generalizes a well known concept of a circuit isomorphism of graphs due to H. Whitney. If has a “dual nature” with respect to ?₂(G ₂) then the concept of (?₁,?₂)-semi-isomorphism of graphs turns into a concept of a (?₁,?₂)-semi-duality of graphs. This gives a natural generalization of the circuit duality of graphs due to H. Whitney. In this paper we investigate (?₁,?₂)-semi-isomorphisms and (?₁,?₂)-semi-dualities of graphs for various families ?₁(G ₁) and ?₂(G ₂). In particular, we consider families of circuits and cocircuits of graphs from this point of view, and obtain some strengthenings of Whitney’s 2-isomorphism theorem and Whitney’s planarity criterion for 3-connected graphs.     

A Note on the Cops and Robber Game on Graphs Embedded in Non-Orientable Surfaces

We consider the two-player, complete information game of Cops and Robber played on undirected, finite, reflexive graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if, after a move, a cop and the robber are on the same vertex. The minimum number of cops needed to catch the robber on a graph is called the cop number of that graph. Let c(g) be the supremum over all cop numbers of graphs embeddable in a closed orientable surface of genus g, and likewise ${\tilde c(g)}$ for non-orientable surfaces. It is known (Andreae, 1986) that, for a fixed surface, the maximum over all cop numbers of graphs embeddable in this surface is finite. More precisely, Quilliot (1985) showed that c(g) ≤ 2g + 3, and Schröder (2001) sharpened this to ${c(g)\le \frac32g + 3}$ . In his paper, Andreae gave the bound ${\tilde c(g) \in O(g)}$ with a weak constant, and posed the question whether a stronger bound can be obtained. Nowakowski & Schröder (1997) obtained ${\tilde c(g) \le 2g+1}$ . In this short note, we show ${\tilde c(g) \leq c(g-1)}$ , for any g ≥ 1. As a corollary, using Schröder’s results, we obtain the following: the maximum cop number of graphs embeddable in the projective plane is 3, the maximum cop number of graphs embeddable in the Klein Bottle is at most 4, ${\tilde c(3) \le 5}$ , and ${\tilde c(g) \le \frac32g + 3/2}$ for all other g.     

Gauss-Bonnet Formula, Finiteness Condition, and Characterizations of Graphs Embedded in Surfaces

Let G be an infinite graph embedded in a closed 2-manifold, such that each open face of the embedding is homeomorphic to an open disk and is bounded by finite number of edges. For each vertex x of G, define the combinatorial curvature

关于可嵌人曲面图的列表(d,1)-全标号问题

图的(d,1)-全标号问题最初是由Havet等人提出的.在本文中,我们考虑了可嵌入曲面图的列表(d,1)-全标号问题,并证明了其列表(d,1)-全标号数不超过△(G)+2d.     

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.     

A Polynomial Invariant of Graphs On Orientable Surfaces

We consider cyclic graphs, that is, graphs with cyclic ordersat the vertices, corresponding to 2-cell embeddings of graphsinto orientable surfaces, or combinatorial maps. We constructa three variable polynomial invariant of these objects, thecyclic graph polynomial, which has many of the useful propertiesof the Tutte polynomial. Although the cyclic graph polynomialgeneralizes the Tutte polynomial, its definition is very different,and it depends on the embedding in an essential way. 2000 MathematicalSubject Classification: 05C10.     

On the Stretch Factor of Randomly Embedded Random Graphs

We consider a random graph $\mathcal{G}(n,p)$ whose vertex set $V,$ of cardinality $n,$ has been randomly embedded in the unit square and whose edges, which occur independently with probability $p,$ are given weight equal to the geometric distance between their end vertices. Then each pair $\{u,v\}$ of vertices has a distance in the weighted graph, and a Euclidean distance. The stretch factor of the embedded graph is defined as the maximum ratio of these two distances, over all $\{u,v\}\subseteq V.$ We give upper and lower bounds on the stretch factor (holding asymptotically almost surely), and show that for $p$ not too close to 0 or 1, these bounds are the best possible in a certain sense. Our results imply that the stretch factor is bounded with probability tending to 1 if and only if $n(1-p)$ tends to 0, answering a question of O’Rourke.     

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）.     

关于图的符号边全控制数

引入了图的符号边全控制的概念,给出了一个连通图G的符号边全控制数γs′t(G)的下限,确定所有n阶树T的最小符号边全控制数,并刻划了满足γs′t(G)=E(G)的所有连通图G,最后还提出了一个关于γs′t(G)上界的猜想.     

On the Edge Connectivity of Direct Products with Dense Graphs

Let ${\kappa^{\prime}(G)}$ be the edge connectivity of G and G × H the direct product of G and H. Let H be any graph with minimal degree ${\delta(H)>|V(H)|/2}$ . We prove that for any graph ${G, \kappa^{\prime}(G\times H)=\textup{min}\{2\kappa^{\prime}(G)|E(H)|,\delta(G)\delta(H)\}}$ . In addition, the structure of minimum edge cuts is described. As an application, we present a necessary and sufficient condition for G × K _n (n ≥ 3) to be super edge connected.     

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.     

k-方图的邻点可区别无圈边染色

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.图G的邻点可区别无圈边色数记为χ′_^(aa)(G),即图G的一个邻点可区别无圈边染色所用的最少颜色数.通过构造具体染色的方法,给出了一些k-方图的邻点可区别无圈边色数.     

Edge Domination in Graphs of Cubes

The signed edge domination number and the signed total edge domination number of a graph are considered; they are variants of the domination number and the total domination number. Some upper bounds for them are found in the case of the n-dimensional cube Q _n.     

Boxicity of Graphs on Surfaces

The boxicity of a graph G = (V, E) is the least integer k for which there exist k interval graphs G _i  = (V, E _i ), 1 ≤ i ≤ k, such that ${E = E_1 \cap \cdots \cap E_k}$ . Scheinerman proved in 1984 that outerplanar graphs have boxicity at most two and Thomassen proved in 1986 that planar graphs have boxicity at most three. In this note we prove that the boxicity of toroidal graphs is at most 7, and that the boxicity of graphs embeddable in a surface Σ of genus g is at most 5g + 3. This result yields improved bounds on the dimension of the adjacency poset of graphs on surfaces.