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.

Conflict-free Incidence Coloring of Outer-1-planar Graphs

An incidence of a graph G is a vertex-edge pair(v,e)such that v is incidence with e.A conflict-free incidence coloring of a graph is a coloring of the incidences in such a way that two incidences(u,e)and(v,f)get distinct colors if and only if they conflict each other,i.e.,(i)u=v,(ii)uv is e or f,or(iii)there is a vertex w such that uw=e and vw=f.The minimum number of colors used among all conflict-free incidence colorings of a graph is the conflict-free incidence chromatic number.A graph is outer-1-planar if it can be drawn in the plane so that vertices are on the outer-boundary and each edge is crossed at most once.In this paper,we show that the conflict-free incidence chromatic number of an outer-1-planar graph with maximum degree△is either 2△or 2△+1 unless the graph is a cycle on three vertices,and moreover,all outer-1-planar graphs with conflict-free incidence chromatic number 2△or 2△+1 are completely characterized.An efficient algorithm for constructing an optimal conflict-free incidence coloring of a connected outer-1-planar graph is 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.

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.

List edge and list total coloring of 1-planar graphs

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that each 1-planar graph with maximum degree △ is （A ＋ 1）-edge-choosable and （△ ＋ 2）- total-choosable if △ ≥ 16, and is A-edge-choosable and （△ ＋ 1）-total-ehoosable if △ ≥21. The second conclusion confirms the list coloring conjecture for the class of 1-planar graphs with large maximum degree.

An Improved Upper Bound on the Linear 2-arboricity of 1-planar Graphs

The linear 2-arboricity la2(G) of a graph G is the least integer k such that G can be partitioned into k edge-disjoint forests,whose component trees are paths of length at most 2.In this paper,we prove that if G is a 1-planar graph with maximum degree Δ,then la_(2)(G)≤[(Δ+1)/2]+7.This improves a known result of Liu et al.(2019) that every 1-planar graph G has la_(2)(G)≤[(Δ+1)/2]+14.We also observe that there exists a 7-regular 1-planar graph G such that la2(G)=6=[(Δ+1)/2]+2,which implies that our solution is within 6 from optimal.

Light Subgraphs in the Family of 1-Planar Graphs with High Minimum Degree

A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that each 1-planar graph with minimum degree 7 contains a copy of K2 V （K1 ∪ K2） with all vertices of degree at most 12. In addition, we also prove the existence of a graph K1 V （K1∪K2） with relatively small degree vertices in 1-planar graphs with minimum degree at least 6.

Acyclic Edge Coloring of Triangle-free 1-planar Graphs

A proper edge coloring of a graph G is acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by X＇a（G）, is the least number of colors such that G has an acyclic edge coloring. A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that X＇a（G） ≤△ A（G）＋ 22, if G is a triangle-free 1-planar graph.

Equitable Coloring of Three Classes of 1-planar Graphs

A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. A plane graph with near-independent crossings or independent crossings, say NIC-planar graph or IC-planar graph, is a 1-planar graph with the restriction that for any two crossings the four crossed edges are incident with at most one common vertex or no common vertices, respectively. In this paper, we prove that each 1-planar graph, NIC-planar graph or IC-planar graph with maximum degree A at least 15, 13 or 12 has an equitable △-coloring, respectively. This verifies the well-known Chen-Lih-Wu Conjecture for three classes of 1-planar graphs and improves some known results.

A Totally(Δ+1)-colorable 1-planar Graph with Girth at Least Five

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we prove that every 1-planar graph G with maximum degree △（G） 〉 12 and girth at least five is totally （△（G）＋1）-colorable.

Linear Arboricity of Outer-1-Planar Graphs

A graph is outer-1-planar if it can be drawn in the plane so that all vertices are on the outer face and each edge is crossed at most once.Zhang et al.(Edge covering pseudo-outerplanar graphs with forests,Discrete Math 312:2788-2799,2012;MR2945171)proved that the linear arboricity of every outer-1-planar graph with maximum degree△is exactly[△/2] provided that△=3or△≥5 and claimed that there are outer-1-planar graphs with maximum degree △=4 and linear arboricity[[(O+1)/2]=3.It is shown in this paper that the linear arboricity of every outer-1-planar graph with maximum degree 4 is exactly 2 provided that it admits an outer-1-planar drawing with crossing distance at least 1 and crossing width at least 2,and moreover,none of the above constraints on the crossing distance and Crossing width can be removed..Besides,a polynomial-time algorithm for constructing a path-2-coloring(i.e.,an edge 2-coloring such that each color class induces a linear forest,a disjoint union of paths)of such an outer-1-planar drawing is given.

On Total Colorings of Some Special 1-planar Graphs

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we verify the total coloring conjecture for every 1-planar graph G if either △（G） ≥9 and g（G）≥ 4, or △（G） ≥ 7 and g（G）≥5, where △（G） is the maximum degree of G and g（G） is the girth of G.

List Edge Coloring of Outer-1-planar Graphs

A graph is outer-1-planar if it can be drawn in the plane so that all vertices are on the outer face and each edge is crossed at most once.It is known that the list edge chromatic numberχ′l(G)of any outer-1-planar graph G with maximum degreeΔ(G)≥5 is exactly its maximum degree.In this paper,we proveχ′l(G)=Δ(G)for outer-1-planar graphs G withΔ(G)=4 and with the crossing distance being at least 3.

Joins of 1-Planar Graphs

A graph is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once.In this paper,we study 1-planar graph joins.We prove that the join G ＋ H is 1-planar if and only if the pair [G,H] is subgraph-majorized by one of pairs [C3 ∪ C3,C3],[C4,C4],[C4,C3],[K2,1,1,P3] in the case when both elements of the graph join have at least three vertices.If one element has at most two vertices,then we give several necessary/sufficient conditions for the bigger element.

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.

1-planar Graphs without 4-cycles or 5-cycles are 5-colorable

A graph is 1-planar if it can be drawn on the Euclidean plane so that each edge is crossed by at most one other edge.A proper vertex k-coloring of a graph G is defined as a vertex coloring from a set of k colors such that no two adjacent vertices have the same color.A graph that can be assigned a proper k-coloring is k-colorable.A cycle is a path of edges and vertices wherein a vertex is reachable from itself.A cycle contains k vertices and k edges is a k-cycle.In this paper,it is proved that 1-planar graphs without 4-cycles or 5-cycles are 5-colorable.

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。

<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.

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.