极大平面图的点面全色数

平面图 G(V,E,F)的点面全色数 xs(G)是使得集合 V(G)U F(G)中相邻和相关联的元素均染为不同颜色的最少颜色数.本文证明了:(1)若 G 为极大平面图,则4≤xs(G)≤6;且 xs(G)=4当且仅当 G 为点次模3-正则图.(2)若 G 为△(G)≤3的简单平面图,则 xs(G)≤6.一、引言本文限于考虑平面图 G(V,E,F),其中 V,E,F 分别为 G 的点集合,边集合和面集     

随机图的独立数和分数匹配数的一个注记（英文）

设I为图G顶点集的子集.如果I中的任意两个点均不相邻,则称I为G的独立集.G的最大独立集的阶数称为独立数,记为α(G).图G的分数匹配是边集上的函数f∈[0,1],使得对每个顶点v都有∑f(e)≤1,这里是对所有与顶点v相关联边的函数值求和.分数匹配数β(G)是所有的分数匹配f中∑₍e∈E(G))f(e)的最大值.本文给出了随机图上关于独立数α(G)与分数匹配数β(G)的一些结果.     

图的匹配与拉普拉斯特征值

设G=(V(G),E(G))是一个图,M是E(G)的—个子集.如果M中任意两条边均无公共端点,则称M为图G的匹配.如果图G的一个匹配M中的边恰好关联G的每一个顶点,则称M为图G的完美匹配.如果图G中除了一个顶点以外,其他所有顶点都与匹配M中的边相关联,则称M为图G的几乎完美匹配.如果对任意v∈V(G), G-v均有完美匹配,则称G是因子临界的.本文中,我们给出了判定一个图有完美匹配、或者几乎完美匹配或者是因子临界的拉普拉斯谱条件.     

（mg＋k,mf—k）—图中正交于r个不相交子图的边不变的（g,f）—因子

设m,k和r为正整数,且使l≤k＜m.设G是一个具有顶点集合V(G)和边集合E(G)的图,并设g和f是定义在V(G)上的使对每个x∈V(G)有r≤g(x)≤f(x)的整数值函数.设H1,H2,…,Hr是G的r个顶点不相交的子图且|E(Hi)|=k,1≤i≤r.本文证明了每个(mg+k,mf-k)-图有k个边不相交的(g,f)-因子正交于Hi,1≤i≤r.     

3-一致超图的反馈数研究

超图H=(V,E)顶点集为V,边集为E.S■V是H的顶点子集,如果H/S不含有圈,则称S是H的点反馈数,记τc(H)是H的最小点反馈数.本文证明了:(i)如果H是线性3-一致超图,边数为m,则τc(H)≤m/3;(ii)如果H是3-一致超图,边数为m,则τc(H)≤m/2并且等式成立当且仅当H任何一个连通分支是孤立顶点或者长度为2的圈.A■V是H的边子集,如果H\A不含有圈,则称A是H的边反馈数,记τc′(H)是H的最小边反馈数.本文证明了如果H是含有p个连通分支的3-一致超图,则τc’(H)≤2m-n+p.     

3-一致超图的反馈数研究（英文）

超图H=(V,E)顶点集为V,边集为E.S■V是H的顶点子集,如果H/S不含有圈,则称S是H的点反馈数,记τ_c(H)是H的最小点反馈数.本文证明了:(i)如果H是线性3-一致超图,边数为m,则τ_c(H)≤m/3;(ii)如果H是3-一致超图,边数为m,则τ_c(H)≤m/2并且等式成立当且仅当H任何一个连通分支是孤立顶点或者长度为2的圈.A■V是H的边子集,如果H\A不含有圈,则称A是H的边反馈数,记τ_c′(H)是H的最小边反馈数.本文证明了如果H是含有p个连通分支的3-一致超图,则τ_c'(H)≤2m-n+p.     

树的彩虹控制数的一个多项式时间算法

设G是—个边染色图,G的彩虹子图是所有边都染不同颜色的子图.覆盖V(G)的不相交彩虹星的集合称为彩虹控制星集,图G最小彩虹控制星集的大小称为彩虹控制数,记为γ(G).本文给出了—个在边染色树T上寻找最小彩虹控制星集从而得到T的彩虹控制数的多项式时间算法.     

mC4的顶点被多重色集合可区别的一般边染色北大核心CSCD

简单图G的一个一般边染色是指若干种颜色关于图G的所有边的一个分配,不要求相邻的边被分配不同的颜色.设f是G的使用了k种颜色的一般边染色,若对(?)u,v∈V(G),u≠v,都有与u关联的边的颜色构成的多重集合异于与v关联的边的颜色构成的多重集合,那么称f是使用了k种颜色的顶点被多重色集合可区别的一般边染色.对G进行顶点被多重色集合可区别的一般边染色所需的最少颜色数记为c(G),并且称c(G)为图G的顶点被多重色集合可区别的一般边色数.本文确定了m个C_4的点不交的并mC_4的顶点被多重色集合可区别的一般边色数.     

图的半强积的邻点可区别染色

两个简单图G与H的半强积G·H是具有顶点集V(G)×V(H)的简单图,其中两个顶点(u,v)与(u',v')相邻当且仅当u=u'且vv'∈E(H),或uu'∈E(G)且vv'∈E(H).图的邻点可区别边(全)染色是指相邻点具有不同色集的正常边(全)染色.统称图的邻点可区别边染色与邻点可区别全染色为图的邻点可区别染色.图G的邻点可区别染色所需的最少的颜色数称为邻点可区别染色数,并记为X_a~((r))(G),其中r=1,2,且X_a~((1))(G)与X_a~((2))(G)分别表示G的邻点可区别的边色数与全色数.给出了两个简单图的半强积的邻点可区别染色数的一个上界,并证明了该上界是可达的.然后,讨论了两个树的不同半强积具有相同邻点可区别染色数的充分必要条件.另外,确定了一类图与完全图的半强积的邻点可区别染色数的精确值.     

对于X-join不可分解的无三角形的图

本文中的图都是有限简单图.仅含一个点的图叫作平凡图,不含边的图叫作空图.V(G)与 E(G)分别表示图 G 的点的集合与边的集合.有时以 G 代替 V(G),以 x∈G代替 x∈V(G).对 x∈G,N_G(x)={y∈G|xy∈E(G)}叫作 x 的邻域.下面的概念是 Sabidussi 引入的:令{G_x|x∈X}是图的一个族,指标取自另一个图 X.令#表示 X 中的邻接关系,⊥_x表示 G_x 中的邻接关系,则这一族图的 X-join 是指图 G,G=(?)(G_x×{x}),且 G 中的邻接关系⊥定义为:对 G 中任两个点(a,r)与(b,s),(a,r)⊥(b,s)当且仅当 r#s 或r=s 且 a⊥_rb.     

Optimally super-edge-connected transitive graphs

Let X=(V,E) be a connected regular graph. X is said to be super-edge-connected if every minimum edge cut of X is a set of edges incident with some vertex. The restricted edge connectivity λ′(X) of X is the minimum number of edges whose removal disconnects X into non-trivial components. A super-edge-connected k-regular graph is said to be optimally super-edge-connected if its restricted edge connectivity attains the maximum 2k−2. In this paper, we define the λ′-atoms of graphs with respect to restricted edge connectivity and show that if X is a k-regular k-edge-connected graph whose λ′-atoms have size at least 3, then any two distinct λ′-atoms are disjoint. Using this property, we characterize the super-edge-connected or optimally super-edge-connected transitive graphs and Cayley graphs. In particular, we classify the optimally super-edge-connected quasiminimal Cayley graphs and Cayley graphs of diameter 2. As a consequence, we show that almost all Cayley graphs are optimally super-edge-connected.     

Biased partitions and judicious <Emphasis Type="Italic">k</Emphasis>-partitions of graphs

Let G = (V,E) be a graph with m edges. For reals p ∈ [0, 1] and q = 1- p, let m_p(G) be the minimum of qe(V₁) +pe(V₂) over partitions V = V₁ ∪ V₂, where e(V_i) denotes the number of edges spanned by V_i. We show that if mp(G) = pqm-δ, then there exists a bipartition V₁, V₂ of G such that e(V₁) ≤ p²m - δ + p√m/2 + o(√m) and e(V₂) ≤ q²m - δ + q √m/2 + o(√m) for δ = o(m^2/3). This is sharp for complete graphs up to the error term o(√m). For an integer k ≥ 2, let fk(G) denote the maximum number of edges in a k-partite subgraph of G. We prove that if fk(G) = (1 - 1/k)m + α, then G admits a k-partition such that each vertex class spans at most m/k² - Ω(m/k^7.5) edges for α = Ω(m/k⁶). Both of the above improve the results of Bollobás and Scott.     

The complexity of irredundant sets parameterized by size

An irredundant set of vertices V′V in a graph G=(V,E) has the property that for every vertex uV′, N[V′−{u}] is a proper subset of N[V′]. We investigate the parameterized complexity of determining whether a graph has an irredundant set of size k, where k is the parameter. The interest of this problem is that while most “k-element vertex set” problems are NP-complete, several are known to be fixed-parameter tractable, and others are hard for various levels of the parameterized complexity hierarchy. Complexity classification of vertex set problems in this framework has proved to be both more interesting and more difficult. We prove that the k-element irredundant set problem is complete for W[1], and thus has the same parameterized complexity as the problem of determining whether a graph has a k-clique. We also show that the “parametric dual” problem of determining whether a graph has an irredundant set of size n−k is fixed-parameter tractable.     

A note on G-intersecting families

Consider a graph G and a k-uniform hypergraph on common vertex set [n]. We say that is G-intersecting if for every pair of edges in there are vertices xX and yY such that x=y or x and y are joined by an edge in G. This notion was introduced by Bohman, Frieze, Ruszinkó and Thoma who proved a natural generalization of the Erd s–Ko–Rado Theorem for G-intersecting k-uniform hypergraphs for G sparse and k=O(n^1/4). In this note, we extend this result to .     

Algorithmic complexity of list colorings

Given a graph G = (V,E) and a finite set L(v) at each vertex v ε V, the List Coloring problem asks whether there exists a function f:V → _vεVL(V) such that (i) f(v)εL(v) for each vεV and (ii) f(u) ≠f(v) whenever u, vεV and uvεE. One of our results states that this decision problem remains NP-complete even if all of the followingconditions are met: (1) each set L(v) has at most three elements, (2) each “color” xε_vεVL(v) occurs in at most three sets L(v), (3) each vertex vεV has degree at most three, and (4) G is a planar graph. On the other hand, strengthening any of the assumptions (1)–(3) yields a polynomially solvable problem. The connection between List Coloring and Boolean Satisfiability is discussed, too.     

Primitive graphs with given exponents and minimum number of edges

A graph G = (V, E) on n vertices is primitive if there is a positive integer k such that for each pair of vertices u, v of G, there is a walk of length k from u to v. The minimum value of such an integer, k, is the exponent, exp(G), of G. In this paper, we find the minimum number, h(n, k), of edges of a simple graph G on n vertices with exponent k, and we characterize all graphs which have h(n, k) edges when k is 3 or even.     

List colourings of planar graphs

A graph G = G(V, E) is called L-list colourable if there is a vertex colouring of G in which the colour assigned to a vertex v is chosen from a list L(v) associated with this vertex. We say G is k-choosable if all lists L(v) have the cardinality k and G is L-list colourable for all possible assignments of such lists. There are two classical conjectures from Erd s, Rubin and Taylor 1979 about the choosability of planar graphs:

every planar graph is 5-choosable and,

there are planar graphs which are not 4-choosable.

We will prove the second conjecture.     

Class Steiner trees and VLSI-design

We consider a generalized version of the Steiner problem in graphs, motivated by the wire routing phase in physical VLSI design: given a connected, undirected distance graph with required classes of vertices and Steiner vertices, find a shortest connected subgraph containing at least one vertex of each required class. We show that this problem is NP-hard, even if there are no Steiner vertices and the graph is a tree. Moreover, the same complexity result holds if the input class Steiner graph additionally is embedded in a unit grid, if each vertex has degree at most three, and each class consists of no more than three vertices. For similar restricted versions, we prove MAX SNP-hardness and we show that there exists no polynomial-time approximation algorithm with a constant bound on the relative error, unless P = NP. We propose two efficient heuristics computing different approximate solutions in time O(¦E¦+¦V¦log¦V¦) and in time O(c(¦E¦+¦V¦log¦V¦)), respectively, where E is the set of edges in the given graph, V is the set of vertices, and c is the number of classes. We present some promising implementation results. kw]Steiner Tree; Heuristic; Approximation complexity; MAX-SNP-hardness     

Magic Labeling of Disjoint Union Graphs

Let G be a graph with vertex set V (G), edge set E(G) and maximum degree Δ respectively. G is called degree-magic if it admits a labelling of the edges by integers {1, 2, …,|E(G)|} such that for any vertex v the sum of the labels of the edges incident with v is equal to (1+|E(G)|)/2·d(v), where d(v) is the degree of v. Let f be a proper edge coloring of G such that for each vertex v ∈ V (G),|{e:e ∈ E_v, f(e) ≤ Δ/2}|=|{e:e ∈ E_v, f(e) > Δ/2}|, and such an f is called a balanced edge coloring of G. In this paper, we show that if G is a supermagic even graph with a balanced edge coloring and m ≥ 1, then (2m + 1)G is a supermagic graph. If G is a d-magic even graph with a balanced edge coloring and n ≥ 2, then nG is a d-magic graph. Results in this paper generalise some known results.     

Choosability of planar graphs

A graph G = G(V, E) with lists L(v), associated with its vertices v V, is called L-list colourable if there is a proper vertex colouring of G in which the colour assigned to a vertex v is chosen from L(v). We say G is k-choosable if there is at least one L-list colouring for every possible list assignment L with L(v) = k v V(G).

Now, let an arbitrary vertex v of G be coloured with an arbitrary colour f of L(v). We investigate whether the colouring of v can be continued to an L-list colouring of the whole graph. G is called free k-choosable if such an L-list colouring exists for every list assignment L (L(v) = k v V(G)), every vertex v and every colour f L(v). We prove the equivalence of the well-known conjecture of Erd s et al. (1979): “Every planar graph is 5-choosable” with the following conjecture: “Every planar graph is free 5-choosable”.