3-正则无爪图的导出匹配覆盖(英文)

The induced matching cover number of a graph G without isolated vertices, denoted by imc（G）,is the minimum integer k such that G has k induced matchings {M1,M2,···,Mk}such that,V（M1）∪V（M2）∪···∪V（Mk）covers V（G）.This paper shows that,if G is a 3-regular claw-free graph,then imc（G）∈{2,3}.     

NEIGHBORHOOD UNION OF INDEPENDENT SETS AND HAMILTONICITY OF CLAW-FREE GRAPHS

Let G be a graph,for any u∈V(G),let N(u) denote the neighborhood of u and d(u)=|N(u)| be the degree of u. For any U V(G) ,let N(U)=Uu,∈UN(u), and d(U)=|N(U)|.A graph G is called claw-free if it has no induced subgraph isomorphic to K1.3. One of the fundamental results concerning cycles in claw-free graphs is due to Tian Feng,et al. : Let G be a 2-connected claw-free graph of order n,and d(u) d(v) d(w)≥n-2 for every independent vertex set {u,v,w} of G, then G is Hamiltonian. It is proved that, for any three positive integers s ,t and w,such that if G is a (s t w-1)connected claw-free graph of order n,and d(S) d(T) d(W)＞n-(s t w) for every three disjoint independent vertex sets S,T,W with |S |=s, |T|=t, |W|=w,and S∪T∪W is also independent ,then G is Hamiltonian. Other related results are obtained too.     

Resolvability in circulant graphs

A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v ∈ V (G) there is a vertex w ∈ W such that d(u, w) ≠ d(v, w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim(G). For a vertex u of G and a subset S of V (G), the distance between u and S is the number min s∈S d(u, s). A k-partition Π = {S 1 , S 2 , . . . , S k } of V (G) is called a resolving partition if for every two distinct vertices u, v ∈ V (G) there is a set S i in Π such that d(u, Si )≠ d(v, Si ). The minimum k for which there is a resolving k-partition of V (G) is called the partition dimension of G, denoted by pd(G). The circulant graph is a graph with vertex set Zn , an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i-j (mod n) ∈ C , where CZn has the property that C =-C and 0 ■ C. The circulant graph is denoted by Xn, Δ where Δ = |C|. In this paper, we study the metric dimension of a family of circulant graphs Xn, 3 with connection set C = {1, n/2 , n-1} and prove that dim(Xn, 3 ) is independent of choice of n by showing that dim(Xn, 3 ) ={3 for all n ≡ 0 (mod 4), 4 for all n ≡ 2 (mod 4). We also study the partition dimension of a family of circulant graphs Xn,4 with connection set C = {±1, ±2} and prove that pd(Xn, 4 ) is independent of choice of n and show that pd(X5,4 ) = 5 and pd(Xn,4 ) ={3 for all odd n ≥ 9, 4 for all even n ≥ 6 and n = 7.     

k-fold coloring of planar graphs

A k-fold n-coloring of G is a mapping φ: V (G) → Zk(n) where Zk(n) is the collection of all ksubsets of {1,2,...,n} such that φ(u) ∩φ(v) = φ if uv ∈ E(G).If G has a k-fold n-coloring,i.e.,G is k-fold n-colorable.Let the smallest integer n such that G is k-fold n-colorable be the k-th chromatic number,denoted by χk(G).In this paper,we show that any outerplanar graph is k-fold 2k-colorable or k-fold χk(C*)-colorable,where C* is a shortest odd cycle of G.Moreover,we investigate that every planar graph with odd girth at least 10k-9(k 3) can be k-fold (2k + 1)-colorable.     

图的邻点可区别全色数的一个上界

Let G = （V, E） be a simple connected graph, and ｜V（G）｜ ≥ 2. Let f be a mapping from V（G） ∪ E（G） to {1,2…, k}. If arbitary uv ∈ E（G）,f（u） ≠ f（v）,f（u） ≠ f（uv）,f（v） ≠ f（uv）; arbitary uv, uw ∈ E（G）（v ≠ w）, f（uv） ≠ f（uw）;arbitary uv ∈ E（G） and u ≠ v, C（u） ≠ C（v）, where
C（u）={f（u）}∪{f（uv）｜uv∈E（G）}.
Then f is called a k-adjacent-vertex-distinguishing-proper-total coloring of the graph G（k-AVDTC of G for short）. The number min{k｜k-AVDTC of G} is called the adjacent vertex-distinguishing total chromatic number and denoted by χat（G）. In this paper we prove that if △（G） is at least a particular constant and δ ≥32√△ln△, then χat（G） ≤ △（G） ＋ 10^26 ＋ 2√△ln△.     

Acyclic 6-choosability of Planar Graphs without 5-cycles and Adjacent 4-cycles

A proper vertex coloring of a graph is acyclic if every cycle uses at least three colors. A graph G is acyclically k-choosable if for any list assignment L = {L(v) : v ∈ V(G)} with |L(v)| ≥ k for all v ∈ V(G), there exists a proper acyclic vertex coloring φ of G such that φ(v) ∈ L(v) for all v ∈ V(G). In this paper, we prove that if G is a planar graph and contains no 5-cycles and no adjacent 4-cycles, then G is acyclically 6-choosable.     

中心对称本原矩阵的k点指数集

Let G = (V, E) be a primitive digraph. The vertex exponent of G at a vertex v ∈ V, denoted by expG(v), is the least integer p such that there is a v → u walk of length p for each u ∈ V. We choose to order the vertices of G in the k-point exponent of G and is denoted by expG(k), 1 ≤ k ≤ n. We define the k-point exponent set E(n, k) := {expG(k)| G = G(A) with A ∈ CSP(n)}, where CSP(n) is the set of all n × n central symmetric primitive matrices and G(A) is the associated graph of the matrix A. In this paper, we describe E(n,k) for all n, k with 1 ≤ k ≤ n except n ≡ 1(mod 2) and 1 ≤ k ≤ n - 4. We also characterize the extremal graphs when k = 1.     

树的L(3,2,1)-标号问题

An L(3,2,1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all non-negative integers(labels) such that |f(u)-f(v)|≥3 if d(u,v)=1,|f(u)-f(v)≥2 if d(u,v)=2 and |f(u)-f(v)|≥1 if d(u,v)=3.For a non-negative integer k,a k-L(3,2,1)-labeling is an L(3,2,1)-labeling such that no label is greater than k.The L(3,2,1)-labeling number of G,denoted by λ_(3,2,1)(G), is the smallest number k such that G has a k-L(3,2,1)-labeling.In this article,we characterize the L(3,2,1)-labeling numbers of trees with diameter at most 6.     

任意长圈覆盖指定的独立的点

An invariant σ2（G） of a graph is defined as follows： σ2（G） ：= min{d（u） ＋ d（v）｜u, v ∈V（G）,uv ∈ E（G）,u ≠ v} is the minimum degree sum of nonadjacent vertices （when G is a complete graph, we define σ2（G） = ∞）. Let k, s be integers with k ≥ 2 and s ≥ 4, G be a graph of order n sufficiently large compared with s and k. We show that if σ2（G） ≥ n ＋ k- 1, then for any set of k independent vertices v1,..., vk, G has k vertex-disjoint cycles C1,..., Ck such that ｜Ci｜ ≤ s and vi ∈ V（Ci） for all 1 ≤ i ≤ k.
The condition of degree sum σs（G） ≥ n ＋ k - 1 is sharp.     

Every Graph Embedded on the Surface with Euler Characteristic Number ε =-1 is Acyclically 11-choosable

A proper vertex coloring of a graph G is acyclic if there is no bicolored cycles in G. A graph G is acyclically k-choosable if for any list assignment L = {L(v) : v ∈ V(G)} with |L(v)| ≥ k for each vertex v ∈ V(G), there exists an acyclic proper vertex coloring φ of G such that φ(v) ∈ L(v)for each vertex v ∈ V(G). In this paper, we prove that every graph G embedded on the surface with Euler characteristic number ε =-1 is acyclically 11-choosable.     

关于乘积Sombor指数的极值(分子)树

拓扑指数是一类可以用来预测化合物的物理化学性质的数值不变量, 其并被广泛用于量子化学、分子生物学和其他研究领域. 对于一个顶点集为$V(G)$、边集为$E(G)$的(分子)图$G$, 其Sombor指数定义为$SO(G)=\sum\limits_{uv\in E(G)}\sqrt{d_{G}^{2}(u)+d_{G}^{2}(v)}$, 其中$d_{G}(u)$表示顶点$u$在$G$中的度. 相应地, 乘积Sombor指数定义为$\prod\nolimits_{SO}(G)= \prod\limits_{uv\in E(G)}\sqrt{d_{G}^{2}(u)+d_{G}^{2}(v)}$. 分子树是最大度$\Delta\leq 4$的树. 在本文中, 我们首先确定了乘积Sombor指数最大的分子树, 然后我们确定了乘积Sombor指数的前十三小的(分子)树.     

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.     

环状硅酸盐图的友好指数集

Let G be a graph with vertex set V(G) and edge set E(G). A labeling f : V(G) →Z2 induces an edge labeling f*: E(G) → Z2 defined by f*(xy) = f(x) + f(y), for each edge xy ∈ E(G). For i ∈ Z2, let vf(i) = |{v ∈ V(G) : f(v) = i}| and ef(i) = |{e ∈ E(G) : f*(e) =i}|. A labeling f of a graph G is said to be friendly if |vf(0)- vf(1)| ≤ 1. The friendly index set of the graph G, denoted FI(G), is defined as {|ef(0)- ef(1)|: the vertex labeling f is friendly}. This is a generalization of graph cordiality. We investigate the friendly index sets of cyclic silicates CS(n, m).     

一个关于平面图的K-(2,1)-全可选性的结果

设图$G$的一个列表分配为映射$L: V(G)\bigcup E(G)\rightarrow2^{N}$. 如果存在函数$c$使得对任意$x\in V(G)\cup E(G)$有$c(x)\in L(x)$满足当$uv\in E(G)$时, $|c(u)-c(v)|\geq1$, 当边$e_{1}$和$e_{2}$相邻时, $|c(e_{1})-c(e_{2})|\geq1$, 当点$v$和边$e$相关联时, $|c(v)-c(e)|\geq 2$, 则称图$G$为$L$-$(p,1)$-全可标号的. 如果对于任意一个满足$|L(x)|=k,x\in V(G)\cup E(G)$的列表分配$L$来说, $G$都是$L$-$(2,1)$-全可标号的, 则称$G$是 $k$-(2,1)-全可选的. 我们称使得$G$为$k$-$(2,1)$-全可选的最小的$k$为$G$的$(2,1)$-全选择数, 记作$C_{2,1}^{T}(G)$. 本文, 我们证明了若$G$是一个$\Delta(G)\geq 11$的平面图, 则$C_{2,1}^{T}(G)\leq\Delta+4$.     

Zeros of Monomial Brauer Characters

Let G be a finite group and p be a fixed prime. A p-Brauer character of G is said to be monomial if it is induced from a linear p-Brauer character of some subgroup(not necessarily proper) of G. Denote by IBr_m(G) the set of irreducible monomial p-Brauer′characters of G. Let H = G′O~p′(G) be the smallest normal subgroup such that G/H is an abelian p′-group. Suppose that g ∈ G is a p-regular element and the order of gH in the factor group G/H does not divide |IBr_m(G)|. Then there exists ? ∈ IBr_m(G) such that ?(g) = 0.     

单圈图的最大Balaban指数与最大和Balaban指数

The Balaban index of a connected graph G is defined as J(G) =|E(G)|μ + 1∑e=uv∈E(G)1√DG(u)DG(v),and the Sum-Balaban index is defined as SJ(G) =|E(G)|μ + 1∑e=uv∈E(G)1√DG(u)+DG(v),where DG(u) =∑w∈V(G)dG(u, w), and μ is the cyclomatic number of G. In this paper, the unicyclic graphs with the maximum Balaban index and the maximum Sum-Balaban index among all unicyclic graphs on n vertices are characterized, respectively.     

An Existence Result for a Class of Chemically Reacting Systems with Sign-Changing Weights

We prove the existence of positive solutions for the system$$\begin{align*}\begin{cases}-\Delta_{p} u =\lambda a(x){f(v)}{u^{-\alpha}},\qquad x\in \Omega,\\-\Delta_{q} v = \lambda b(x){g(u)}{v^{-\beta}},\qquad x\in \Omega,\\u = v =0, \qquad x\in\partial \Omega,\end{cases}\end{align*}$$where $\Delta_{r}z={\rm div}(|\nabla z|^{r-2}\nabla z)$, for $r>1$ denotes the r-Laplacian operator and $\lambda$ is a positive parameter, $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, $n\geq1$ with sufficiently smooth boundary and $\alpha, \beta \in (0,1).$ Here $ a(x)$ and $ b(x)$ are $C^{1}$ sign-changingfunctions that maybe negative near the boundary and $f,g $ are $C^{1}$ nondecreasing functions, such that $f, g :\ [0,\infty)\to [0,\infty);$ $f(s)>0,$ $g(s)>0$ for $s> 0$, $\lim_{s\to\infty}g(s)=\infty$ and$$\lim_{s\to\infty}\frac{f(Mg(s)^{\frac{1}{q-1}})}{s^{p-1+\alpha}}=0,\qquad \forall M>0.$$We discuss the existence of positive weak solutions when $f$, $g$, $a(x)$ and $b(x)$ satisfy certain additional conditions. We employ the method of sub-supersolution to obtain our results.     

L7（3）与GL7（3）的OD-刻画

对于任意一个有限群G,令π（G）表示由它的阶的所有素因子构成的集合.构建一种与之相关的简单图,称之为素图,记作Γ（G）.该图的顶点集合是π（G）,图中两顶点p,g相连（记作p～q）的充要条件是群G恰有pq阶元.设π（G）={P₁,p2,…,p_x}.对于任意给定的p∈π（G）,令deg（p）:=|{q∈π（G）|在素图Γ（G）中,p～q}|,并称之为顶点p的度数.同时,定义D（G）:=（deg（p₁）,deg（p₂）,…,deg（p_s））,其中p₁2<…相似文献   

广义超特殊p-群的自同构群Ⅲ

确定了广义超特殊p-群G的自同构群的结构.设|G|=p~(2n+m),|■G|=p~m,其中n≥1,m≥2,Aut_fG是AutG中平凡地作用在Frat G上的元素形成的正规子群,则(1)当G的幂指数是p~m时,(i)如果p是奇素数,那么AutG/AutfG≌Z_((p-1)p~(m-2)),并且AutfG/InnG≌Sp(2n,p)×Zp.(ii)如果p=2,那么AutG=Aut_fG(若m=2)或者AutG/AutfG≌Z_(2~(m-3))×Z_2(若m≥3),并且AutfG/InnG≌Sp(2n,2)×Z_2.(2)当G的幂指数是p~(m+1)时,(i)如果p是奇素数,那么AutG=〈θ〉■Aut_fG,其中θ的阶是(p-1)p~(m-1),且Aut_f G/Inn G≌K■Sp(2n-2,p),其中K是p~(2n-1)阶超特殊p-群.(ii)如果p=2,那么AutG=〈θ_1,θ_2〉■Aut_fG,其中〈θ_1,θ_2〉=〈θ_1〉×〈θ_2〉≌Z_(2~(m-2))×Z_2,并且Aut_fG/Inn G≌K×Sp(2n-2,2),其中K是2~(2n-1)阶初等Abel 2-群.特别地,当n=1时...