On the (Δ + 2)-Total-Colorability of Planar Graphs with 7-Cycles Containing at Most Two Chords

The Total Coloring Conjecture (TCC) proposes that every simple graph G is (Δ + 2)-totally-colorable, where Δ is the maximum degree of G. For planar graph, TCC is open only in case Δ = 6. In this paper, we prove that TCC holds for planar graph with Δ = 6 and every 7-cycle contains at most two chords.

Perfect 1-k Matchings of Bipartite Graphs

Let k be a positive integer and G a bipartite graph with bipartition (X,Y). A perfect 1-k matching is an edge subset M of G such that each vertex in Y is incident with exactly one edge in M and each vertex in X is incident with exactly k edges in M. A perfect 1-k matching is an optimal semi-matching related to the load-balancing problem, where a semi-matching is an edge subset M such that each vertex in Y is incident with exactly one edge in M, and a vertex in X can be incident with an arbitrary number of edges in M. In this paper, we give three sufficient and necessary conditions for the existence of perfect 1-k matchings and for the existence of 1-k matchings covering | X |−dvertices in X, respectively, and characterize k-elementary bipartite graph which is a graph such that the subgraph induced by all k-allowed edges is connected, where an edge is k-allowed if it is contained in a perfect 1-k matching.

Edge Colorings of Planar Graphs without 6-Cycles with Two Chords

It is proved here that if a planar graph has maximum degree at least 6 and any 6-cycle contains at most one chord, then it is of class 1.

4-Cycle Decompositions of Graphs

In this paper we consider the problem of finding the smallest number such that any graph G of order n admits a decomposition into edge disjoint copies of C4 and single edges with at most elements. We solve this problem for n sufficiently large.

Bondage Number of 1-Planar Graph

The bondage number of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph a domination number greater than the domination number of G. In this paper, we prove that for a 1-planar graph G.

<i>L</i>(0, 1)-Labelling of Cactus Graphs

An L(0,1)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between the labels assigned to any two adjacent vertices is at least zero and the difference between the labels assigned to any two vertices which are at distance two is at least one. The span of an L(0,1)-labelling is the maximum label number assigned to any vertex of G. The L(0,1)-labelling number of a graph G, denoted by λ0.1(G) is the least integer k such that G has an L(0,1)-labelling of span k. This labelling has an application to a computer code assignment problem. The task is to assign integer control codes to a network of computer stations with distance restrictions. A cactus graph is a connected graph in which every block is either an edge or a cycle. In this paper, we label the vertices of a cactus graph by L(0,1)-labelling and have shown that, △-1≤λ0.1(G)≤△ for a cactus graph, where △ is the degree of the graph G.

A Note on SK, SK₁, SK₂Indices of Interval Weighted Graphs

In this study, the SK, SK1 and SK2 indices are defined on weighted graphs. Then, the SK, SK1 and SK2 indices are defined on interval weighted graphs. Their behaviors are investigated under some graph operations by using these definitions.

<i>L</i>(2,1)-Labeling of the Brick Product Graphs

A k-L(2,1)-labeling for a graph G is a function such that whenever and whenever u and v are at distance two apart. The λ-number for G, denoted by λ(G), is the minimum k over all k-L(2,1)-labelings of G. In this paper, we show that for or 11, which confirms Conjecture 6.1 stated in [X. Li, V. Mak-Hau, S. Zhou, The L(2,1)-labelling problem for cubic Cayley graphs on dihedral groups, J. Comb. Optim. (2013) 25: 716-736] in the case when or 11. Moreover, we show that? if 1) either (mod 6), m is odd, r = 3, or 2) (mod 3), m is even (mod 2), r = 0.

Intergenic subset organization within a set of geographically-defined viral sequences from the 2009 H1N1 influenza A pandemic

We report a bioinformatic analysis of the datasets of sequences of all ten genes from the 2009 H1N1 influenza A pandemic in the state of Wisconsin. The gene with the greatest summed information entropy was found to be the hemagglutinin (HA) gene. Based upon the viral ID identifier of the HA gene sequence, the sequences of all of the genes were sorted into two subsets, depending upon whether the nucleotide occupying the position of maximum entropy, position 658 of the HA sequence, was either A or U. It was found that the information entropy (H) distributions of subsets differed significantly from each other, from H distributions of randomly generated subsets and from the H distributions of the complete datasets of each gene. Mutual information (MI) values facilitated identification of nine nucleotide positions, distributed over seven of the influenza genes, at which the nucleotide subsets were disjoint, or almost disjoint. Nucleotide frequencies at these nine positions were used to compute mutual information values that subsequently served as weighting factors for edges in a graph net-work. Seven of the nucleotide positions in the graph network are sites of synonymous mutations. Three of these sites of synonymous mutation are within a single gene, the M1 gene, which occupied the position of greatest graph centrality. It is proposed that these bioinformatic and network graph results may reflect alterations in M1-mediated viral packaging and exteriorization, known to be susceptible to synonymous mutations.

Markov Chains Based on Random Generalized 1-Flipper Operations for Connected Regular Multi-digraphs

The properties of generalized flip Markov chains on connected regular digraphs are discussed.The 1-Flipper operation on Markov chains for undirected graphs is generalized to that for multi-digraphs.The generalized 1-Flipper operation preserves the regularity and weak connectivity of multi-digraphs.The generalized 1-Flipper operation is proved to be symmetric.Moreover,it is presented that a series of random generalized 1-Flipper operations eventually lead to a uniform probability distribution over all connected d-regular multi-digraphs without loops.

K_(1,k)-FACTORIZATION OF BIPARTITE GRAPHS

In this paper, a necessary condition for a bipartite graph λK m,n to be K 1,k factorizable and a sufficient condition for kK m,n to have a K 1,k factorization whenever k is a prime number are given.

Acyclic Edge Coloring of 1-planar Graphs without 4-cycles

An acyclic edge coloring of a graph G is a proper edge coloring such that there are no bichromatic cycles in G.The acyclic chromatic index χ'α(G) of G is the smallest k such that G has an acyclic edge coloring using k colors.It was conjectured that every simple graph G with maximum degree Δ has χ'_α(G) ≤Δ+2.A1-planar graph is a graph that can be drawn in the plane so that each edge is crossed by at most one other edge.In this paper,we show that every 1-planar graph G without 4-cycles has χ'_α(G)≤Δ+22.

血清Nrf2、Keap1水平变化与糖尿病视网膜病变分期间关系

目的探讨血清单核细胞核因子E2相关因子(Nrf2)、Kelch样ECH关联蛋白1(Keap1)水平变化与糖尿病视网膜病变(DR)分期间关系。方法选取2020年4月—2023年2月如皋市中医院DR患者147例作为观察组,并根据DR分期分为非增生型糖尿病视网膜病变(NPDR)、增生型糖尿病视网膜病变(PDR);另选取同期糖尿病无视网膜病变患者49例作为对照Ⅰ组,健康体检者49例作为对照Ⅱ组。对比3组及观察组不同DR分期患者血清Nrf2、Keap1水平,Spearman秩相关系数分析血清Nrf2、Keap1水平与DR分期间关系;多因素logistic回归分析血清Nrf2、Keap1水平对DR分期的影响;限制性立方样条图分析血清Nrf2、Keap1水平与DR分期的剂量-效应关系。结果观察组血清Nrf2水平低于对照Ⅰ组、对照Ⅱ组,Keap1水平高于对照Ⅰ组、对照Ⅱ组,对照Ⅰ组血清Nrf2水平低于对照Ⅱ组,Keap1水平高于对照Ⅱ组(P<0.05);随DR分期增加,血清Nrf2水平呈降低趋势,血清Keap1水平呈升高趋势(P<0.05);血清Nrf2水平与DR分期呈负相关(r=-0.806,P<0.001),血清Keap1水平与DR分期呈正相关(r=0.854,P<0.001);经多因素logistic回归分析,高血压、糖尿病发病年龄、总胆固醇对DR分期无显著影响(P>0.05),血清Nrf2水平是DR分期的独立保护因素,高脂血症、糖尿病病程、空腹血糖、糖化血红蛋白、血清Keap1水平是DR分期的独立危险因素(P<0.05);限制性立方样条图分析显示,血清Nrf2(χ^(2)=11.800,P<0.001)、Keap1(χ^(2)=8.401,P=0.015)水平与DR分期间存在非线性关系,控制其他病变为固定变量,血清Nrf2水平<1.70μg/L、血清Keap1水平>3.00μg/L时,PDR风险显著增加。结论血清Nrf2、Keap1水平变化与DR分期关系密切,血清Nrf2水平<1.70μg/L、血清Keap1水平>3.00μg/L时提示DR进展至PDR的风险显著增加。

1-平面图的线性荫度(英文)

证明了最大度△≥33的1-平面图的线性荫度为[△/2]

图的L(d,1,1)-标号

给出了图L(d,1,1)-标号的一般性质.对一般图G,给出了构造L(d,1,1)-标号的一个算法,证明了λd,1,1(G)≤Δ3-Δ2+dΔ.对最大度Δ的树T,证明了d+Δ-1≤λd,1,1(T)≤d+2Δ-2,并且式中的上界与下界都是可达的.此外,对于两类特殊的树图:拟正则树TΔ及正则毛毛虫Catn,给出了确切的L(d,1,1)-标号数,其中d≥2.

具有交换点稳定子群的6度1-正则Cayley图

令Γ是一个图,如果Γ的图自同构群Aut(Γ)作用在其弧集上正则,则称图Γ为1-正则图。本文给出具有交换点稳定子群的6度1-正则Cayley图的一个完全分类,证明了在同构意义下具有交换点稳定子群的无核6度1-正则Cayley图只有一个。

边色临界图的1-因子和几乎1-因子的存在性

根据Vizing邻接引理和关于临界图的独立数的一个结论,利用图的1-因子和几乎1-因子存在的充要条件,采用结构图论的方法证明了:1)若G是2n阶Δ-临界图,且Δ≥n,δ≥n-2,则G存在1-因子;2)若G是2n+1阶Δ-临界图,且Δ≥n+1,δ≥n-2,则G存在几乎1-因子.

具有初等交换点稳定子的9度1-正则Cayley图

对于一个图Γ,如果它的图自同构群Aut(Γ)作用在它的弧集上正则,则称图Γ为1-正则图。本文给出了具有初等交换点稳定子的9度1-正则Cayley图的一个完全分类,证明了在同构意义下,具有初等交换点稳定子的9度无核1-正则Cayley图只有一个。

点稳定子为Z_4×Z_2的8度1-正则Cayley图

一个图Γ称为1-正则图,如果图Γ的图自同构群Aut(Γ)作用在它的弧集上正则.本文给出了点稳定子为Z4×Z2的8度1-正则Cayley图的一个完全分类。

Hamiltonian[k,k+1]-因子(英文)

本文考虑n/2-临界图中Hamiltonian[k,k+1]-因子的存在性。Hamiltonian[k,k+1]-因子是指包含Hamiltonian圈的[k,k+1]-因子;给定阶数为n的简单图G,若δ(G)≥n/2而δ(G\e)<n/2(对任意的e∈E(G)),则称G为n/2-临界图。设k为大于等于2的整数,G为n/2-临界图(其中n≥4k-6且n≥7),我们证明了对于G的任何Hamiltonian圈C,G中必存在包含C的[k,k+1]-因子。该结果改进了现有的一些有关Hamiltonian[k,k+1]-因子存在性的结果。