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Injective coloring of planar graphs with girth 5

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Frontiers of Mathematics in China 2022年第3期17卷 473-484页

作者： Yuehua BU Piaopiao YE School of Mathematical Sciences Zhejiang Normal UniversityJinhua 321004China Xingzhi College Zhejiang Normal UniversityJinhua 321004China

A coloring of a graph G is injective if its restriction to the neighbour of any vertex is injective.The injective chromatic number x_(i)(G)of a graph G is the least k such that there is an injective k-coloring.In this... 详细信息

关键词： Planar graph girth injective coloring face

Partitioning Planar Graphs with girth at Least 6 into Bounded Size Components

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Journal of Mathematical Research with Applications 2023年第1期43卷 16-24页

作者： Chunyu TIAN Lei SUN School of Mathematics and Statistics Shandong Normal University

An(Ok1、Ok2)-partition of a graph G is the partition of V(G) into two non-empty subsets V1and V2,such that G[V1] and G[V2] are graphs with components of order at most k1and k2,respectively.In this paper,we consider the problem of partitioning the vertex set of a planar graph with girth restriction such that each part induces a graph with components of bounded order.We prove that every planar graph with girth at least 6 and i-cycle is not intersecting with j-cycle admits an(O2,O3)-partition,where i∈{6,7,8} and j∈{6,7,8,9}.

关键词： planar graph face girth vertex partition discharging procedure

Some Explicit Constructions of Type-Ⅱ,Ⅲ,Ⅳ,Ⅴ QC-LDPC Codes with girth 6

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China Communications 2020年第5期17卷 89-109页

作者： Farzaneh Abedi Mohammad Gholami Department of Mathematics Shahrekord UniversityP.O.Box 88186-34141ShahrekordIran School of Computer Sciences Institute for Research in Fundamental Sciences(IPM)P.O.Box 19395-5746TehranIran

Multi-type quasi-cyclic(QC) low-density parity-check(LDPC) codes can be considered as multiple-edge protograph QC-LDPC codes having some advantages in the minimum Hamming distance bound over single-edge protograph codes or type-Ⅰ QC-LDPC codes when the base matrices have the same size. In this paper, we investigate a class of multi-type QC-LDPC codes whose parity-check matrices contain just one blockrow of circulants and we obtain the generator matrix of such codes in general form. Using the permutation arrays and defining injection arrays, we present a new approach to construct a class of high-rate type-Ⅰ QC-LDPC codes with girth 6 from the constructed 4-cycle free multi-type QC-LDPC codes. In continue, for 2 ≤ w≤6, some type-w QC-LDPC codes with girth 6 are constructed explicitly such that the constructed codes are flexible in terms of rate and length. To the best of our knowledge, for w = 5,6, this is the first paper which deals with the explicit construction of type-w QC-LDPC codes with girth 6 and high rates. Moreover, for w = 3, 4, the constructed type-w QC-LDPC codes have better(6,8)-cycle multiplicities than the codes with minimum achievable length recently constructed by cyclic difference families(CDFs). Simulation results show that the binary and non-binary constructed codes outperform the constituent underlying QC-LDPC codes.

关键词： QC-LDPC codes permutation arrays girth explicit constructions

Combinatorial design-based quasi-cyclic LDPC codes with girth eight

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Digital Communications and Networks 2018年第4期4卷 296-300页

作者： Sina Vail Narges Rezvani Majid College of Ertgineering Information Technology and Environment Charles Din-win UmversityEllengowan DriveNT0909Australia

This paper presents a novel regular Quasi-Cyclic (QC)Low Density Parity Check (LDPC)codes with columnweight three and girth at least eight.These are designed on the basis of combinatorial design in which subsets applied for the construction of circulant matrices are determined by a particular subset.Considering the nonexistence of cycles four and six in the structure of the parity check matrix,a bound for their minimum weight is proposed.The simtdations conducted confirm that without applying a masking technique,the newly implemented codes have a performance similar to or better than other well-known codes.This is evident in the waterfall region, while their error floor at very low Bit Error Rate (BER)is expected.

关键词： Quasi-cyclic LDPC codes Combinatorial design Minimum weight girth

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

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Acta Mathematica Sinica,English Series 2016年第11期32卷 1337-1349页

作者： Lin SUN Jian Liang WU Hua CAI School of Mathematics Shandong University Ji'nan 250100 P. R. China Department of Mathematics Changji University Changji 831100 P. R. China

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... 详细信息

关键词： 1-planar graph total coloring discharging method girth

Relationship between heart girth, serum progesterone and superovulation response ofdonor Holstein cows

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Asian pacific Journal of Reproduction 2019年第3期8卷 118-123页

作者： Abdel-Tawab AYKhalil Ahmed Abdel-Wahab Rabie LAbdel Aziz Department of Theriogenology Faculty of Veterinary MedicineBeni-Suef UniversityBeni-SuefEgypt Physiology Department Faculty of Veterinary MedicineMinia UniversityEl-MiniaEgypt

Objective: To determine effects of variations of heart girth, volume trait on embryo quality grade, and to explore the relationship between heart girth and circulating progesterone and correlations of circulating progesterone with embryo quality grade in superstimulated donor Holstein cows. Methods: Nineteen cows were subjected to a standard superstimulation protocol using follicle stimulating hormone. Blood samples were collected before superovulation, at insemination and at collection of embryos for progesterone analysis. Embryo quality grades were compared between high and low heart girth donors. Moreover, Pearson's correlations were determined between heart girth, progesterone and embryo quality. Results: Variation of heart girth was not associated with significant differences in embryo quality grade between high and low heart girth donor cows (P>0.05). However, we observed a significantly higher percentage of the third-grade embryo (24.09%) in low heart girth donors compared to 9.64% in high heart girth donors. Moreover, the percentage of the transferable embryo was numerically higher and that of the degenerated embryo was numerically lower in low heart girth cows. Donor cows with low response to superovulation (total structures ≤ 3) expressed numerically higher mean heart girth, compared to donors with high (total structures>15) and medium (total structures = 4-15) superovulation. Heart girth and body weight of donor cows were moderately correlated (r=0.45, Pgirth may influence the response of Holstein cows to superovulation;however, future studies with a higher number of cows are warranted to clarify significant inf

关键词： Heart girth Superovulation Embryo quality Progesterone Holstein cow

On the girth of Tanner (5,7) Quasi-Cyclic LDPC Codes

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Journal of Harbin Institute of Technology(New Series) 2017年第6期24卷 80-89页

作者： Hengzhou Xu Baoming Bai Dan Feng Cheng Sun State Key Laboratory of Integrated Services Networks Xidian UniversityXi’an 710071China

The girth plays an important role in the design of LDPC codes. In order to determine the girth of Tanner(5,7) quasi-cyclic( QC) LDPC codes with length 7p for p being a prime with the form 35 m + 1,the cycles of lengths 4,6,8,and 10 are analyzed. Then these cycles are classified into sixteen categories,each of which can be expressed as an ordered block sequence,or a certain type. It is also shown that the existence of these cycles is equal to polynomial equations over Fpwho has a 35th unit root. We check if these polynomial equations have a 35th unit root and obtain the girth values of Tanner(5,7) QC LDPC codes.

关键词： LDPC codes quasi-cyclic Tanner graph girth

Ruling out small stopping sets and small girth in Tanner graph of QC-LDPC code

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Journal of Systems Engineering and Electronics 2010年第1期21卷 134-137页

作者： Lingjun Kong Yang Xiao Institute of Information Science Beijing Jiaotong University Beijing 100044 E R. China

The existing constructions of quasi-cyclic low-density parity-check （QC-LDPC） codes do not consider the problems of small stopping sets and small girth together in the Tanner graph, while their existences will lead to the bit error rate （BER） performance of QC-LDPC codes being much poorer than that of randomly constructed LDPC codes even decoding failure. To solve the problem, some theorems of the specific chosen parity-check matrix of QC-LDPC codes without small stopping sets and small girth are proposed. A novel construction for QC-LDPC codes with long block lengths is presented by multiplying mmin or the multiple of mmin, which is the minimum order of the identity matrix for the chosen parity-check matrix. The simulation results show that the specific chosen parity-check matrix of QC-LDPC codes can effectively avoid specified stopping sets and small girth and exhibit excellent BER performance than random LDPC codes with the same longer codes length.

关键词： quasi-cyclic low-density parity-check codes stoppingset stopping distance girth circulant matrices.

Design of good QC-LDPC codes without small girth in the p-plane

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Journal of Systems Engineering and Electronics 2011年第2期22卷 183-187页

作者： Lingjun Kong Yang Xiao Institute of Information Science Beijing Jiaotong University Beijing 100044 E R. China Faculty of Information Engineering and Automation Kunming University of Science and Technology Kunming 650500 P. R. China

A construction method based on the p-plane to design high-girth quasi-cyclic low-density parity-check （QC-LDPC） codes is proposed. Firstly the good points in every line of the p-plane can be ascertained through filtering the bad points, because the designed parity-check matrixes using these points have the short cycles in Tanner graph of codes. Then one of the best points from the residual good points of every line in the p-plane will be found, respectively. The optimal point is also singled out according to the bit error rate （BER） performance of the QC-LDPC codes at last. Explicit necessary and sufficient conditions for the QC-LDPC codes to have no short cycles are presented which are in favor of removing the bad points in the p-plane. Since preventing the short cycles also prevents the small stopping sets, the proposed construction method also leads to QC-LDPC codes with a higher stopping distance.

关键词： quasi-cyclic low-density parity-check （QC-LDPC）codes circulant matrices girth stopping set stopping distance.