带松弛条件的图的强边着色

给定两个非负整数s和t,图G的(s,t)-松弛强k边着色可表示为映射c：E(G)→[k],这个映射满足对G中的任意一条边e,颜色c(e)在e的1-邻域中最多出现s次并且在e的2-邻域中最多出现t次。图G的(s,t)-松弛强边着色指数,记作χ'_(s,t)(G),表示使得图G有(s,t)-松弛强k边着色的最小k值。在图G中,如果mad(G) < 3并且Δ≤4,那么χ'_(1,0)(G)≤3Δ。并证明如果G是平面图,最大度Δ≥4并且围长最少为7,那么χ'_(1,0)(G)≤3Δ-1。     

关于拟k-连通图的一个注释

G是一个k-连通图,T是G的一个k-点割,若G-T可被划分成两个子图G₁,G₂,且|G₁|≥2,|G₂|≥2,则称T是G的一个非平凡点割。假定G是一个不含非平凡（k-1）点割的（k-1）-连通图,则称G是一个拟k-连通图。证明了对任意一个k≥5且t> $ \frac{k}{2}$的整数,若G是一个不含（K₂+tK₁）的k-连通图,且G中任意两个不同点对v,w,有d（v）+d（w）≥ $\frac{{3k}}{2} $+t,则对G中的任意一个点,存在一条与之关联的边收缩后可以得到一个拟k-连通图,且G中至少有$\frac{{\left| {V\left( G \right)} \right|}}{2} $条边使得收缩其中任意一条边后仍是拟k-连通的。     

围长至少为6的平面图的邻点可区别边染色

邻点可区别边染色是指图G有一个正常边染色且任意两个相邻顶点的颜色集合不相等.邻点可区别边色数是指使图G有一个邻点可区别边染色的最小颜色数值,记作χ_α’(G).本文证明了:若图G是围长至少为6的正常平面图,则有χ_α’(G)≤max{6,△(G)+1}.     

外平面图的弱完备染色

假设G=（V,E,F）是一个平面图。如果e₁和e₂是G中两条相邻边且在关联的面的边界上连续出现,那么称e₁和e₂面相邻。图G的一个弱完备k-染色是指存在一个从V ∪ E ∪ F到k色集合{1, …, K}的映射,使得任意两个相邻点,两个相邻面,两条面相邻的边,以及V ∪ E ∪ F中任意两个相关联的元素都染不同的颜色。若图G有一个弱完备k-染色,则称G是弱完备k-可染的。平面图G的弱完备色数是指G是弱完备k-可染的正整数k的最小值,记成χ_vef（G）。2016年,Fabrici等人猜想：每个无环且无割边的连通平面图是弱完备7-可染的。证明外平面图满足猜想,即外平面图是弱完备7-可染的。     

Pm□Pn的Smarandachely-邻点可区别边色数

图G的正常边染色f满足相邻点的色集合相不互包含时,该染色称为图G的Smarandcchely-邻点可区别边染色,其中S(x)={f(xw)|xw∈E(G)}称之为在f下的顶点x的色集合.该染色称为图G的Smarandchely-邻点可区别边染色.对图G进行的.Smarandchely-邻点可区别边染色所用最少颜色数称为图G的Smarandachely-邻点可区别边色数.讨论了Pm□Pn的Smarandchely-邻点可区别边色数.     

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.     

围长至少为4的平面图的邻点可区别边色数

图G的邻点可区别边染色是G的正常边染色,使得每一对相邻顶点有不同的颜色集合．G的邻点可区别边色数X＇a（G）是使得G有一个k-邻点可区别边染色的最小正整数七．本文证明了：若G是围长至少为4且最大度至少为6的平面图,则X＇a（G）≤△＋2．     

路、圈的广义Mycielski图的邻点可区别Ⅰ-全色数

图G的邻点可区别Ⅰ-全染色是一个满足相邻顶点色集合不同的Ⅰ-全染色,其中任意一点的色集合包含该顶点及其关联边所染的颜色.所需颜色的最小数称为邻点可区别Ⅰ-全色数,记作χ_atⁱ(G).研究了路和圈的广义Mycielski图的邻点可区别Ⅰ-全色数:对于阶数n≥2的路P_n,当n=2,3,4时,有χ_atⁱ(M(P_n))=n+1;否则,χ_atⁱ(M(P_n))=n.对于阶数n≥3的圈C_n,当n=3,4时,有χ_atⁱ(M(C_n))=5;否则,χ_atⁱ(M(C_n))=n.     

某些中间图的邻点可区别E-全色数

设G（V,E）是简单连通图,T（G）为图G的所有顶点和边构成的集合,并设C是k-色集（k是正整数）,若T（G）到C的映射f满足：对任意uv∈E（G）,有f（u）≠f（v）,f（u）≠f（uv）,f（v）≠f（uv）,并且C（u）≠C（v）,其中C（u）={f（u）}∪{f（uv）｜uv∈E（G）}.那么称f为图G的邻点可区别E-全染色（简记为k-AVDETC）,并称χ_（at）~e（G）=min{k｜图G有k-邻点可区别E-全染色}为G的邻点可区别E-全色数.图G的中间图M（G）就是在G的每一个边上插入一个新的顶点,再把G上相邻边上的新的顶点相联得到的.探讨了路、圈、扇、星及轮的中间图的邻点可区别E-全染色,并给出了这些中间图的邻点可区别E-全色数.     

图的一般邻点可区别均匀边染色和均匀全染色

提出了一般邻点可区别均匀边染色和全染色的新概念,研究了路P_n、圈C_n、星S_n、扇F_n、轮W_n、完全二部图K_(m,n)、2维平面网格图P_m×P_n的一般邻点可区别均匀边染色和全染色,具体给出这些图的一般邻点可区别均匀边染色和全染色指标.     

A Note on Strong Edge Coloring of Sparse Graphs

A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most 2 receive distinct colors. The strong chromatic index χ'_s(G) of a graph G is the minimum number of colors used in a strong edge coloring of G. In an ordering Q of the vertices of G, the back degree of a vertex x of G in Q is the number of vertices adjacent to x, each of which has smaller index than x in Q. Let G be a graph of maximum degree Δ and maximum average degree at most 2 k. Yang and Zhu [J. Graph Theory, 83, 334–339(2016)] presented an algorithm that produces an ordering of the edges of G in which each edge has back degree at most 4 kΔ-2 k in the square of the line graph of G, implying that χ'_s(G) ≤ 4 kΔ-2 k + 1. In this note, we improve the algorithm of Yang and Zhu by introducing a new procedure dealing with local structures. Our algorithm generates an ordering of the edges of G in which each edge has back degree at most(4 k-1)Δ-2 k in the square of the line graph of G, implying that χ'_s(G) ≤(4 k-1)Δ-2 k + 1.     

On <Emphasis Type="Italic">σ</Emphasis>-semipermutable Subgroups of Finite Groups

Let σ={σ_i|i∈I} be some partition of the set of all primes P, G a finite group and σ(G)={σ_i|σ_i ∩ π(G)≠∅}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member≠1 of H is a Hall σ_i-subgroup of G for some σ_i ∈ σ and H contains exactly one Hall σ_i-subgroup of G for every σ_i ∈ σ(G). A subgroup H of G is said to be:σ-semipermutable in G with respect to H if HH_i^x=H_i^xH for all x ∈ G and all H_i ∈ H such that (|H|,|H_i|)=1; σ-semipermutable in G if H is σ-semipermutable in G with respect to some complete Hall σ-set of G. We study the structure of G being based on the assumption that some subgroups of G are σ-semipermutable in G.     

Bandwidth of the composition of two graphs

The bandwidth B(G) of a graph G is the minimum of the quantity max{|f(x)−f(y)| : xyE(G)} taken over all proper numberings f of G. The composition of two graphs G and H, written as G[H], is the graph with vertex set V(G)×V(H) and with (u₁,v₁) is adjacent to (u₂,v₂) if either u₁ is adjacent to u₂ in G or u₁=u₂ and v₁ is adjacent to v₂ in H. In this paper, we investigate the bandwidth of the composition of two graphs. Let G be a connected graph. We denote the diameter of G by D(G). For two distinct vertices x,yV(G), we define w_G(x,y) as the maximum number of internally vertex-disjoint (x,y)-paths whose lengths are the distance between x and y. We define w(G) as the minimum of w_G(x,y) over all pairs of vertices x,y of G with the distance between x and y is equal to D(G). Let G be a non-complete connected graph and let H be any graph. Among other results, we prove that if |V(G)|=B(G)D(G)−w(G)+2, then B(G[H])=(B(G)+1)|V(H)|−1. Moreover, we show that this result determines the bandwidth of the composition of some classes of graphs composed with any graph.     

On the planarity of jump graphs

For a graph G of size m1 and edge-induced subgraphs F and H of size k (1km), the subgraph H is said to be obtained from F by an edge jump if there exist four distinct vertices u,v,w, and x in G such that uvE(F), wxE(G)−E(F), and H=F−uv+wx. The minimum number of edge jumps required to transform F into H is the k-jump distance from F to H. For a graph G of size m1 and an integer k with 1km, the k-jump graph J_k(G) is that graph whose vertices correspond to the edge-induced subgraphs of size k of G and where two vertices of J_k(G) are adjacent if and only if the k-jump distance between the corresponding subgraphs is 1. All connected graphs G for which J₂(G) is planar are determined.     

Multicolored forests in complete bipartite graphs

If the edges of a graph G are colored using k colors, we consider the color distribution for this coloring a=(a₁,a₂,…,a_k), in which a_i denotes the number of edges of color i for i=1,2,…,k. We find inequalities and majorization conditions on color distributions of the complete bipartite graph K_n,n which guarantee the existence of multicolored subgraphs: in particular, multicolored forests and trees. We end with a conjecture on partitions of K_n,n into multicolored trees.     

Stability of Periodic Orbits and Return Trajectories of Continuous Multi-valued Maps on Intervals

Let I be a compact interval of real axis R, and(I, H) be the metric space of all nonempty closed subintervals of I with the Hausdorff metric H and f : I → I be a continuous multi-valued map. Assume that Pn =(x_0, x_1,..., xn) is a return tra jectory of f and that p ∈ [min Pn, max Pn] with p ∈ f(p). In this paper, we show that if there exist k(≥ 1) centripetal point pairs of f(relative to p)in {(x_i; x_i+1) : 0 ≤ i ≤ n-1} and n = sk + r(0 ≤ r ≤ k-1), then f has an R-periodic orbit, where R = s + 1 if s is even, and R = s if s is odd and r = 0, and R = s + 2 if s is odd and r 0. Besides,we also study stability of periodic orbits of continuous multi-valued maps from I to I.     

Solution to an extremal problem on bigraphic pairs with a <Emphasis Type="Italic">Z</Emphasis><Subscript>3</Subscript>-connected realization

Let S=(a₁,...,a_m; b₁,...,b_n), where a₁,...,a_m and b₁,...,b_n are two nonincreasing sequences of nonnegative integers. The pair S=(a₁,...,a_m; b₁,...,b_n) is said to be a bigraphic pair if there is a simple bipartite graph G=(X ∪ Y, E) such that a₁,...,a_m and b₁,...,b_n are the degrees of the vertices in X and Y, respectively. Let Z₃ be the cyclic group of order 3. Define σ(Z₃, m, n) to be the minimum integer k such that every bigraphic pair S=(a₁,...,a_m; b₁,...,b_n) with a_m, b_n ≥ 2 and σ(S)=a₁ +... + a_m ≥ k has a Z₃-connected realization. For n=m, Yin[Discrete Math., 339, 2018-2026 (2016)] recently determined the values of σ(Z₃, m, m) for m ≥ 4. In this paper, we completely determine the values of σ(Z₃, m, n) for m ≥ n ≥ 4.     

The smallest degree sum that yields potentially kCℓ-graphic sequences

Gould et al. (Combinatorics, Graph Theory and Algorithms, Vol. 1, 1999, pp. 387–400) considered a variation of the classical Turán-type extremal problems as follows: For a given graph H, determine the smallest even integer σ(H,n) such that every n-term graphic sequence π=(d₁,d₂,…,d_n) with term sum σ(π)=d₁+d₂++d_nσ(H,n) has a realization G containing H as a subgraph. In this paper, for given integers k and ℓ, ℓ7 and 3kℓ, we completely determine the smallest even integer σ(_kC_ℓ,n) such that each n-term graphic sequence π=(d₁,d₂,…,d_n) with term sum σ(π)=d₁+d₂++d_nσ(_kC_ℓ,n) has a realization G containing a cycle of length r for each r, krℓ.     

The connectivity of a graph on uniform points on [0,1]

A random graph G_n(x) is constructed on independent random points U₁,…,U_n distributed uniformly on [0,1]^d, d1, in which two distinct such points are joined by an edge if the l_∞-distance between them is at most some prescribed value 0<x<1. The connectivity distance c_n, the smallest x for which G_n(x) is connected, is shown to satisfy

(1)

For d2, the random graph G_n(x) behaves like a d-dimensional version of the random graphs of Erdös and Rényi, despite the fact that its edges are not independent: c_n/d_n→1, a.s., as n→∞, where d_n is the largest nearest-neighbor link, the smallest x for which G_n(x) has no isolated vertices.     

A note on linearization of actions of finitely semisimple Hopf algebras on local algebras

Let H be a Hopf algebra over a field k and let H A → A, h a → h.a, be an action of H on a commutative local Noetherian kalgebra (A, m). We say that this action is linearizable if there exists a minimal system x₁, …, x_n of generators of the maximal ideal m such that h.x_i ε kx₁ + …+ kx_n for all h ε H and i = 1, …, n. In the paper we prove that the actions from a certain class are linearizable (see Theorem 4), and we indicate some consequences of this fact.