Disjoint Paths in Graphs II,A Special Case

Let G be a graph,{a,b,c} í V(G) \{a,b,c\}\subseteq V(G) , and {a￠,b￠,c￠} í V(G) \{a',b',c'\}\subseteq V(G) such that {a,b,c} 1 {a￠,b￠,c￠} \{a,b,c\}\neq \{a',b',c'\} . We say that (G,{a,c}, {a￠,c￠}, (b, b￠)) (G,\{a,c\}, \{a',c'\}, (b, b')) is an obstruction if, for any three vertex disjoint paths from {a, b, c} to {a', b', c'} in G, one path is from b to b'. In this paper, we characterize a special class of obstructions. This will be used to characterize all obstructions.     

Disjoint Paths in Graphs I,¶3-planar Graphs and Basic Obstructions

Let G be a graph,{a,b,c} í V(G) \{a,b,c\}\subseteq V(G) , and {a￠,b￠,c￠} í V(G) \{a',b',c'\}\subseteq V(G) such that {a,b,c} 1 {a￠,b￠,c￠} \{a,b,c\}\neq \{a',b',c'\} . We say that (G,{a,c}, {a￠,c￠}, (b, b￠)) (G,\{a,c\}, \{a',c'\}, (b, b')) is an obstruction if, for any three vertex disjoint paths from {a, b, c} to {a', b', c'} in G, one path is from b to b'. Robertson and Seymour asked the problem of characterizing all obstructions. In this paper, we present a list of "basic" obstructions and show how to produce other obstructions from these basic ones. We also prove results about disjoint paths in graphs. Results in this paper will be used in subsequent papers to characterize all obstructions.     

Degree Bounded Spanning Trees

In this paper, we give a sufficient condition for a graph to have a degree bounded spanning tree. Let n ≥ 1, k ≥ 3, c ≥ 0 and G be an n-connected graph. Suppose that for every independent set ${S \subseteq V(G)}In this paper, we give a sufficient condition for a graph to have a degree bounded spanning tree. Let n ≥ 1, k ≥ 3, c ≥ 0 and G be an n-connected graph. Suppose that for every independent set S í V(G){S \subseteq V(G)} of cardinality n(k−1) + c + 2, there exists a vertex set X í S{X \subseteq S} of cardinality k such that the degree sum of vertices in X is at least |V(G)| − c −1. Then G has a spanning tree T with maximum degree at most k+éc/nù{k+\lceil c/n\rceil} and ?_{v ? V(T)}max{d_T(v)-k,0} ￡ c{\sum_{v\in V(T)}\max\{d_T(v)-k,0\}\leq c} .     

2-Reducible Two Paths and an Edge Constructing a Path in (2k + 1)-Edge-Connected Graphs

Let k ≥ 5 be an odd integer and G = (V(G), E(G)) be a k-edge-connected graph. For ${X\subseteq V(G),e(X)}$ denotes the number of edges between X and V(G) ? X. We here prove that if ${\{s_i,t_i\}\subseteq X_i\subseteq V(G)(i=1,2),f}$ is an edge between s ₁ and ${s_2,X_1\cap X_2=\emptyset,e(X_1)\le 2k-3,e(X_2)\le 2k-2}$ , and e(Y) ≥ k + 1 for each ${Y\subseteq V(G)}$ with ${Y\cap\{s_1,t_1,s_2,t_2\}=\{s_1,t_2\}}$ , then there exist paths P ₁ and P ₂ such that P _i joins s _i and ${t_i,V(P_i)\subseteq X_i}$ (i = 1, 2) and ${G-f-E(P_1\cup P_2)}$ is (k ? 2)-edge-connected, and in fact we give a generalization of this result.     

具误差隐格式迭代逼近严格伪压缩映像族公共不动点 (英)

设$K$是实Banach空间$E$中非空闭凸集, $\{T_i\}_i=1^{N}$是$N$个具公共不动点集$F$的严格伪压缩映像, $\{\alpha_n\}\subset [0,1]$是实数列, $\{u_n\}\subset K$是序列, 且满足下面条件 (i)\ 设$K$是实Banach空间$E$中非空闭凸集, $\{T_i\}_i=1^{N}$是$N$个具公共不动点集$F$的严格伪压缩映像, $\{\alpha_n\}\subset [0,1]$是实数列, $\{u_n\}\subset K$是序列, 且满足下面条件 (i)\ 设$K$是实Banach空间$E$中非空闭凸集, $\{T_i\}_i=1^{N}$是$N$个具公共不动点集$F$的严格伪压缩映像, $\{\alpha_n\}\subset [0,1]$是实数列, $\{u_n\}\subset K$是序列, 且满足下面条件 (i)\ 设K是实Banach空间E中非空闭凸集，{Ti}i=1^N是N个具公共不动点集F的严格伪压缩映像，{αn}包括于[0,1]是实数例，{un}包括于K是序列，且满足下面条件（i）0〈α≤αn≤1;（ii）∑n=1∞（1-αn）=＋∞.（iii）∑n=1∞ ‖un‖〈＋∞.设x0∈K,{xn}由正式定义xn=αnxn-1＋（1-αn）Tnxn＋un-1,n≥1,其中Tn=Tnmodn，则下面结论（i）limn→∞‖xn-p‖存在，对所有p∈F；（ii）limn→∞d（xn,F）存在，当d（xn,F）=infp∈F‖xn-p‖；（iii）lim infn→∞‖xn-Tnxn‖=0.文中另一个结果是，如果{xn}包括于[1-2^-n,1]，则{xn}收敛，文中结果改进与扩展了Osilike（2004）最近的结果，证明方法也不同。     

Constructing quasi-random subsets of ZN by using elliptic curves

Let ε:y2 =x3 + Ax + B be an elliptic curve defined over the finite field Zp(p ＞ 3)and G be a rational point of prime order N on ε.Define a subset of ZN,the residue class ring modulo N,as S ∶={n ∶n ∈ZN,...     

单圈图的最大特征值的上界的改进

Let G（V, E） be a unicyclic graph, Cm be a cycle of length m and Cm G, and ui ∈ V（Cm）. The G - E（Cm） are m trees, denoted by Ti, i = 1, 2,..., m. For i = 1, 2,..., m, let eui be the excentricity of ui in Ti and ec = max{eui ： i = 1, 2 , m}. Let κ = ec＋1. Forj = 1,2,...,k- 1, let δij = max{dv ： dist（v, ui） = j,v ∈ Ti}, δj = max{δij ： i = 1, 2,..., m}, δ0 = max{dui ： ui ∈ V（Cm）}. Then λ1（G）≤max{max 2≤j≤k-2 （√δj-1-1＋√δj-1）,2＋√δ0-2,√δ0-2＋√δ1-1}. If G ≌ Cn, then the equality holds, where λ1 （G） is the largest eigenvalue of the adjacency matrix of G.     

半无爪可迹图的度和条件

可迹图即为一个含有Hamilton路的图.令$N[v]=N(v)\cup\{v\}$, $J(u,v)=\{w\in N(u)\cap N(v):N(w)\subseteq N[u]\cup N[v]\}$.若图中任意距离为2的两点$u,v$满足$J(u,v)\neq \emptyset$,则称该图为半无爪图.令$\sigma_{k}(G)=\min\{\sum_{v\in S}d(v):S$为$G$中含有$k$个点的独立集\},其中$d(v)$表示图$G$中顶点$v$的度.本论文证明了若图$G$为一个阶数为$n$的连通半无爪图,且$\sigma_{3}(G)\geq {n-2}$,则图$G$为可迹图; 文中给出一个图例,说明上述结果中的界是下确界; 此外,我们证明了若图$G$为一个阶数为$n$的连通半无爪图,且$\sigma_{2}(G)\geq \frac{2({n-2})}{3}$,则该图为可迹图.     

图$(\bigcup\limits_{i{\in}A}P_i){\bigcup}(\bigcup\limits_{j{\in}B}U_j)$伴随唯一的充要条件

设$h(G; x) =h(G)$和$[G]_h$分别表示图$G$的伴随多项式和伴随等价类. 文中给出了$[G]_h$的一个新应用. 利用$[G]_h$, 给出了图$H{\;}(H \cong G)$伴随唯一的充要条件, 其中$H=(\bigcup_{i{\in}A}P_i){\bigcup}(\bigcup_{j{\in}B}U_j)$, $A \subseteq A^{'}=\{1,2,3,5\} \bigcup \{2n|n \in N, n \geq 3\}$, $B \subseteq B^{'}     

Proof of a conjecture of V. Nikiforov

Using analytical tools, we prove that for any simple graph G on n vertices and its complement [`(G)]\bar G the inequality $\mu \left( G \right) + \mu \left( {\bar G} \right) \leqslant \tfrac{4} {3}n - 1$\mu \left( G \right) + \mu \left( {\bar G} \right) \leqslant \tfrac{4} {3}n - 1 holds, where μ(G) and m( [`(G)] )\mu \left( {\bar G} \right) denote the greatest eigenvalue of adjacency matrix of the graphs G and [`(G)]\bar G respectively.     

Reflection Spaces and Corresponding Kinematic Structures

For a reflection space (P, Γ) [introduced in Karzel and Taherian (Results Math 59:213–218, 2011)] we define the notion “Reducible Subspace”, consider two subsets of ${\Gamma, \Gamma^{+} := \{a b\,|\, a,b \in P\}}$ and ${\Gamma^{-} := \{a b c\,|\, a, b, c \in P\}}$ and the map $$ \kappa : 2^{P} \to 2^{\Gamma^+} ; X \mapsto X \cdot X := \{xy\,|\, x,y \in X\}$$ We show, for each subspace S of (P, Γ), V := κ(S) is a v-subgroup (i.e. V is a subgroup of Γ⁺ with if ${\xi = xy \in V, \xi \neq 1}$ then ${x \cdot \overline{x,y}\subseteq V}$ ) if and only if S is reducible. Our main results are stated in the items 1–5 in the introduction.     

Traces, traceability, and lattices of traces under the set theoretic inclusion

Let a trace be a computably enumerable set of natural numbers such that ${V^{[m]} = \{n : \langle n, m\rangle \in V \}}$ V [ m ] = { n : 〈 n , m 〉 ∈ V } is finite for all m, where ${\langle^{.},^{.}\rangle}$ 〈 . , . 〉 denotes an appropriate pairing function. After looking at some basic properties of traces like that there is no uniform enumeration of all traces, we prove varied results on traceability and variants thereof, where a function ${f : \mathbb{N} \rightarrow \mathbb{N}}$ f : N → N is traceable via a trace V if for all ${m, \langle f(m), m\rangle \in V.}$ m , 〈 f ( m ) , m 〉 ∈ V . Then we turn to lattices $$\textit{\textbf{L}}_{tr}(V) = (\{W : V \subseteq W \, {\rm and} \, W \, {\rm a} \, {\rm trace}\}, \, \subseteq),$$ L t r ( V ) = ( { W : V ? W and W a trace } , ? ) , V a trace. Here, we study the close relationship to ${\mathcal{E} = (\{A : A \subseteq \mathbb{N} \quad c.e.\}, \subseteq)}$ E = ( { A : A ? N c . e . } , ? ) , automorphisms, isomorphisms, and isomorphic embeddings.     

Toughness and Matching Extension in {\mathcal{P}_3}-Dominated Graphs

Let G be a connected graph. For ${x,y\in V(G)}$ with d(x, y) = 2, we define ${J(x,y)= \{u \in N(x)\cap N(y)\mid N[u] \subseteq N[x] \,{\cup}\,N[y] \}}$ and ${J'(x,y)= \{u \in N(x) \cap N(y)\,{\mid}\,{\rm if}\ v \in N(u){\setminus}(N[x] \,{\cup}\, N[y])\ {\rm then}\ N[x] \,{\cup}\, N[y]\,{\cup}\,N[u]{\setminus}\{x,y\}\subseteq N[v]\}}$ . A graph G is quasi-claw-free if ${J(x,y) \not= \emptyset}$ for each pair (x, y) of vertices at distance 2 in G. Broersma and Vumar (in Math Meth Oper Res. doi:10.1007/s00186-008-0260-7) introduced ${\mathcal{P}_{3}}$ -dominated graphs defined as ${J(x,y)\,{\cup}\, J'(x,y)\not= \emptyset}$ for each ${x,y \in V(G)}$ with d(x, y) = 2. This class properly contains that of quasi-claw-free graphs, and hence that of claw-free graphs. In this note, we prove that a 2-connected ${\mathcal{P}_3}$ -dominated graph is 1-tough, with two exceptions: K _2,3 and K _1,1,3, and prove that every even connected ${\mathcal{P}_3}$ -dominated graph ${G\ncong K_{1,3}}$ has a perfect matching. Moreover, we show that every even (2p + 1)-connected ${\mathcal{P}_3}$ -dominated graph is p-extendable. This result follows from a stronger result concerning factor-criticality of ${\mathcal{P}_3}$ -dominated graphs.     

On Bernstein–Walsh-type lemmas in regions of the complex plane

Let G ì \mathbb C G \subset {\mathbb C} be a finite region bounded by a Jordan curve L: = ?G L: = \partial G , let W: = \textext[`(G)] \Omega : = {\text{ext}}\bar{G} (with respect to [`(\mathbb C)] {\overline {\mathbb C}} ), $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} , and let w = F(z) w = \Phi (z) be a univalent conformal mapping of Ω onto Δ normalized by $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 . By A _p (G); p > 0; we denote a class of functions f analytic in G and satisfying the condition

On super vertex-graceful unicyclic graphs

A graph G with p vertices and q edges, vertex set V(G) and edge set E(G), is said to be super vertex-graceful (in short SVG), if there exists a function pair (f, f ⁺) where f is a bijection from V(G) onto P, f ⁺ is a bijection from E(G) onto Q, f ⁺((u, v)) = f(u) + f(v) for any (u, v) ∈ E(G),

关于$\{1,3,4\}$-逆的混合逆序律

In this paper, we study the mixed-type reverse order laws to {1, 3, 4}-inverses for closed range operators A, B and AB. It is shown that B{1, 3, 4}A{1, 3, 4} ?(AB){1, 3} if and only if R(A*AB) ? R(B). For every A(134)∈ A{1, 3, 4}, it has(A(134)AB){1, 3, 4}A{1, 3, 4} =(AB){1, 3, 4} if and only if R(AA*AB) ? R(AB). As an application of our results, some new characterizations of the mixed-type reverse order laws associated to the Moore-Penrose inverse and the {1, 3, 4}-inverse are established.     

Min and max functions,Bochner-Radon-Nikodym approximations,and a representation characterization

Summary By means of basic properties of the functions min{x, y} and max{x, y} andsum supremum and sum infimum functionals, an elementary argument for a nonintegrable form of the Lipschitz portion of the Bochner-Radon-Nikodym Theorem is obtained. It is also shown that if a < b and h is a function from [a; b] into ,then the following two statements are equivalent: 1) there is a set U, a field F of subsets of U, a function from F into exp ([a; b]) and a real nonnegative-valued finitely additive function on F such that if ap 0, if a (h(s)–h(r))/(s–r)> (h(t)–h(s))/(t–)>0     

On 2-factors with given properties in claw-free graphs

Let G be a claw-free graph such that (i) k(G) 3 2k(G) \geq 2, (ii) $|V (G)| \geq 8$|V (G)| \geq 8 and (iii) d(G) 3 4\delta(G) \geq 4. For every pair of edges e₁, e₂ of G the graph G^* = G - {e₁, e₂}G^* = G - \{e_1, e_2\} has a 2-factor.     

A relationship between bounds on the sum of squares of degrees of a graph

LetG = (V, E) be a simple graph withn vertices and e edges. Letdi be the degree of the ith vertex v_i ∈ V andm _i the average of the degrees of the vertices adjacent to vertexv _i ∈ V. It is known by Caen [1] and Das [2] that $\frac{{4e^2 }}{n} \leqslant d_1^2 + ... + d_n^2 \leqslant e max \{ d_j + m_j |v_j \in V\} \leqslant e\left( {\frac{{2e}}{{n - 1}} + n - 2} \right)$ . In general, the equalities do not hold in above inequality. It is shown that a graphG is regular if and only if $\frac{{4e^2 }}{n} = d_1^2 + ... + d_n^2 $ . In fact, it is shown a little bit more strong result that a graphG is regular if and only if $\frac{{4e^2 }}{n} = d_1^2 + ... + d_n^2 = e max \{ d_j + m_j |v_j \in V\} $ . For a graphG withn < 2 vertices, it is shown that G is a complete graphK _n if and only if $\frac{{4e^2 }}{n} = d_1^2 + ... + d_n^2 = e max \{ d_j + m_j |v_j \in V\} = e\left( {\frac{{2e}}{{n - 1}} + n - 2} \right)$ .     

Quadratic forms on graphs

We introduce a new graph parameter, called the Grothendieck constant of a graph G=(V,E), which is defined as the least constant K such that for every A:E→ℝ,