几乎无爪图的叶子数较少的生成树

A spanning tree with no more than 3 leaves is called a spanning 3-ended tree.In this paper, we prove that if G is a k-connected(k ≥ 2) almost claw-free graph of order n and σ_(k+3)(G) ≥ n + k + 2, then G contains a spanning 3-ended tree, where σk(G) =min{∑_(v∈S)deg(v) : S is an independent set of G with |S| = k}.     

关于图中存在不含给定k-因子的[a,b]-因子的度条件

设$1\leq a<b, 0\leq k$是整数. 设$G$是一个含有$k$-因子$Q$且阶为$|G|$的图. 设\delta(G)$表示$G$的最小度, 且$\delta(G)\geq a+k$. 如果$Q$连通, 设$\varepsilon=k$, 否则设$\varepsilon=k+1$.证明:当$b\geq a+\varepsilon-1$时, 如果对$G$的任意两个不相邻的点$x$和$y$都有max$\{d_G(x),d_G(y)\}\geq {\rm max}\{{{a|G|} \over {a+b}},{{(|G|+(a-1)(2a+b+\varepsilon-2))} \over {b+1}}\}+k$, 那么$G$有一个$[a, b]$-因子$F$ 使得 $E(F)\cap E(Q)=\emptyset$. 这个度条件是最佳的, 条件$b\geqa+\varepsilon-1$不能去掉. 进一步,得到图存在含给定$k$-因子的$[a, b]$-因子的度条件.     

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.     

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$C_{2,1}^{T}(G)\leq\Delta+4$.     

笛卡尔乘积图的圈点连通度

图G的圈点连通度,记为κ_c(G),是所有圈点割中最小的数目,其中每个圈点割S满足G-S不连通且至少它的两个分支含圈.这篇文章中给出了两个连通图的笛卡尔乘积的圈点连通度:(1)如果G_1≌K_m且G_2≌K_n,则κ_c(G_1×G_2)=min{3m+n-6,m+3n-6},其中m+n≥8,m≥n+2,或n≥m+2,且κ_c(G_1×G_2)=2m+2n-8,其中m+n≥8,m=n,或n=m+1,或m=n+11;(2)如果G_1≌K_m(m≥3)且G_2■K_n,则min{3m+κ(G_2)-4,m+3κ(G_2)-3,2m+2κ(G_2)-4}≤κ_c(G_1×G_2)≤mκ(G2);(3)如果G_1■K_m,K_(1,m-1)且G_2■K_n,K_(1,n-1),其中m≥4,n≥4,则min{3κ(G_1)+κ(G_2)-1,κ(G_1)+3κ(G_2)-1,2_κ(G_1)+2_κ(G_2)-2}≤κ_c(G_1×G_2)≤min{mκ(G_2),nκ(G_1),2m+2n-8}.     

无$K_{1,4}$子图的图的路覆盖

For a graph G, a path cover is a set of vertex disjoint paths covering all the vertices of G, and a path cover number of G, denoted by p(G), is the minimum number of paths in a path cover among all the path covers of G. In this paper, we prove that if G is a K_(1,4)-free graph of order n and σ_(k+1)(G) ≥ n-k, then p(G) ≤ k, where σ_(k+1)(G) = min{∑v∈S d(v) : S is an independent set of G with |S| = k + 1}.     

Dirac K4细分定理的一种推广

1960年, Dirac证明了对一个阶为$n\geq 4$的图$G$,如果$G$的边数大于$2n-3$,那么$G$一定包含一个$K_4$的细分. 作者证明了对一个阶为$n\geq 4$的图$G$和$k\geq 2$,如果$G$的边数至少为$kn-\frac{(k-1)(k+2)}{2}$, 那么$G$一定包含一个$W_{k+1}$的细分,从而推广了Dirac的结果.另外,作者利用范更华提出的边切换的方法,给出了Dirac结果的另一种证明.     

不含某些子图的k连通图中的k可收缩边

最近Ando等证明了在一个$k$($k\geq 5$ 是一个整数) 连通图 $G$ 中,如果 $\delta(G)\geq k+1$, 并且 $G$ 中既不含 $K^{-}_{5}$,也不含 $5K_{1}+P_{3}$, 则$G$ 中含有一条 $k$ 可收缩边.对此进行了推广,证明了在一个$k$连通图$G$中,如果 $\delta(G)\geq k+1$,并且 $G$ 中既不含$K_{2}+(\lfloor\frac{k-1}{2}\rfloor K_{1}\cup P_{3})$,也不含 $tK_{1}+P_{3}$ ($k,t$都是整数,且$t\geq 3$),则当 $k\geq 4t-7$ 时, $G$ 中含有一条 $k$ 可收缩边.     

三边连通超欧拉图的一个注记

For two integers l 0 and k ≥ 0,define C(l,k) to be the family of 2-edge connected graphs such that a graph G ∈ C(l,k) if and only if for every bond S-E(G) with |S| ≤ 3,each component of G-S has order at least(|V(G)|-k)/l.In this note we prove that if a 3-edge-connected simple graph G is in C(10,3),then G is supereulerian if and only if G cannot be contracted to the Petersen graph.Our result extends an earlier result in [Supereulerian graphs and Petersen graph.JCMCC 1991,9:79-89] by Chen.     

C2(4,k) 中的强支撑可迹图

设2≤h≤3,l0,k≥0是整数,C_h(l,k)是由h-边连通简单图组成的集合,图G∈C_h(l,k)当且仅当对图G的任意一个二边割或三边割X,图G-X的每个分支都至少有︱V(G)-k︱/l个点.设e=u_1v_1和e'=u_2v_2是图G的两条边.若e≠e',G(e,e')是将图G中的边e=u_1v_1和e'=u_2v_2分别用路u_1v_ev_1和u_2v_e'v_2替换得到的图(其中,v_e,v_e'是不在V(G)中的两个新的点).若e=e',G(e,e')是将图G中的边e=u_1v_1用路u_1v_ev_1替换得到的图,也记作G(e).若对任意的e,e'∈E(G),G(e,e')都有支撑(v_e,v_e')迹,则称图G是强支撑可迹的.作者证明了,若图G∈C_2(4,k)且|V(G)|5k,则要么图G是强支撑可迹图,要么存在e,e'∈E(G),使得G(e,e')可以收缩成一个有限图类F中的图.当k=4时,F被完全确定了.     

边临界图的新下界

图$G$ 为简单的第二类连通图, 且对$G$ 的任意边$e$,有$\chi^{\prime}(G-e)<\chi^{\prime}(G)$, 则称 $G$是临界的.该文给出了阶为$n$ 边数为$m$的$\Delta$ -临界图的新下界, 即$m\geq(3\Delta+6)n/10$, 这里$1\leq\Delta\leq18$     

图的距离无符号拉普拉斯谱半径的关于色数的下界

Let G be a connected graph on n vertices with chromatic number k, and let ρ(G)be the distance signless Laplacian spectral radius of G. We show that ρ(G) ≥ 2n + 2「n k」- 4,with equality if and only if G is a regular Tur′an graph.     

关于完全三部图色唯一性的一个注记

Let P(G,λ) be the chromatic polynomial of a simple graph G. A graph G is chromatically unique if for any simple graph H, P(H,λ) = P(G,λ) implies that H is isomorphic to G. Many sufficient conditions guaranteeing that some certain complete tripartite graphs are chromatically unique were obtained by many scholars. Especially, in 2003, Zou Hui-wen showed that if n 31m2 + 31k2 + 31mk+ 31m? 31k+ 32√m2 + k2 + mk, where n,k and m are non-negative integers, then the complete tripartite graph K(n - m,n,n + k) is chromatically unique (or simply χ-unique). In this paper, we prove that for any non-negative integers n,m and k, where m ≥ 2 and k ≥ 0, if n ≥ 31m2 + 31k2 + 31mk + 31m - 31k + 43, then the complete tripartite graph K(n - m,n,n + k) is χ-unique, which is an improvement on Zou Hui-wen's result in the case m ≥ 2 and k ≥ 0. Furthermore, we present a related conjecture.     

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△.     

图的联结数与[a,b]-因子存在性

设G是一个n阶图,a,b,m1,m2是非负整数且满足1≤a＜b和b≥m1.H1和H2是图G的两个边不交的子图且满足|E(H1)|=m1和|E(H2)|=m2.证明下列结论:若图G的联结数bind(G)＞(a+b-1)(n-1)/bn-(a+b)-2(m1+m2)+2且n≥(b-1)(a+b-1)(a+b-2)+2b(m1+m2)/b(b-1),则图G有一个[a,b]-因子F满足E(H1)(∈)E(F)和E(H2)∩ E(F)=φ.进一步指出这个结果是最好的.     

图的上可嵌入性与独立顶点的新度和

Let G be a k(k ≤3)-edge connected simple graph with minimal degree ≥ 3,girth g,r=g12.For any independent set {a1,a2 , . . . , a 6/(4 k)} of G,if,then G is up-embeddable.     

完全图的Cantesian积的L(2,1)-圆标定

For positive integers j and k with j ≥ k, an L（j, k）-labeling of a graph G is an assignment of nonnegative integers to V（G） such that the difference between labels of adjacent vertices is at least j, and the difference between labels of vertices that are distance two apart is at least k. The span of an L（j, k）-labeling of a graph G is the difference between the maximum and minimum integers it uses. The λj, k-number of G is the minimum span taken over all L（j, k）-labelings of G. An m-（j, k）-circular labeling of a graph G is a function f ： V（G） →{0, 1, 2,..., m - 1} such that ｜f（u） - f（v）｜m ≥ j if u and v are adjacent; and ｜f（u） - f（v）｜m 〉 k ifu and v are at distance two, where ｜x｜m = min{｜xl｜, m-｜x｜}. The minimum integer m such that there exists an m-（j, k）-circular labeling of G is called the σj,k-number of G and is denoted by σj,k（G）. This paper determines the σ2,1-number of the Cartesian product of any three complete graphs.     

与直径和围长有关的最大亏格的下界的一个注记

改正了文章"与直径和围长有关的最大亏格的下界(数学学报2004,47(6):1201-1204)"中的一个错误结论,并得到了如下结果:设G是直径为d(G)的简单图,若G的围长g(G)■d(G),则ξ(G)■2,从而γM(G)■(1/2)β(G)-1.     

Halin-图的点强全染色

图G(V,E)的一个k-正常全染色f叫做一个k-点强全染色当且仅当对任意v∈V(G), N[v]中的元素被染不同色,其中N[v]={u|uv∈V(G)}∪{v}．χTvs(G)=min{k|存在图G的k- 点强全染色}叫做图G的点强全色数．对3-连通平面图G(V,E),如果删去面fo边界上的所有点后的图为一个树图,则G(V,E)叫做一个Halin-图．本文确定了最大度不小于6的Halin- 图和一些特殊图的的点强全色数XTvs(G),并提出了如下猜想:设G(V,E)为每一连通分支的阶不小于6的图,则χTvs(G)≤△(G) 2,其中△(G)为图G(V,E)的最大度．     

变更图的直径

P(t,n)和C(t,n)分别表示在阶为n的路和圈中添加t条边后得到的图的最小直径；f(t,k)表示从直径为k的图中删去t条边后得到的连通图的最大直径．这篇文章证明了t≥4且n≥5时,P(t,n)≤(n-8)/(t 1) 3；若t为奇数,则C(t,n)≤(n-8)/(t 1) 3；若t为偶数,则C(t,n)≤(n-7)/(t 2) 3．特别地,「(n-1)/5」≤P(4,n)≤「(n 3)/5」,「n/4」-1≤C(3,n)≤「n/4」．最后,证明了：若k≥3且为奇数,则f(t,k)≥(t 1)k-2t 4．这些改进了某些已知结果．