Funds: The study was supported by the National Natural Science Foundation of China (12071453) and the National Key R&D Program of China (2020YFA0713100).
Zelin Ren is currently a master student under the supervision of Prof. Xinmin Hou at the University of Science and Technology of China. His research mainly focuses on random graphs
Xinmin Hou received his Ph.D. from Dalian University of Technology. He is currently a Professor at the University of Science and Technology of China. His research interests focus on graph theory, combinatorial optimization and cyberspace security
Dirac’s theorem states that if a graph G on n vertices has a minimum degree of at least n2, then G contains a Hamiltonian cycle. Bohman et al. introduced the random perturbed graph model and proved that for any constant α>0 and a graph H with a minimum degree of at least αn, there exists a constant C depending on α such that for any p⩾, H \cup {G_{n,p}} is asymptotically almost surely (a.a.s.) Hamiltonian. In this study, the random perturbed digraph model is considered, and we show that for all \alpha = \omega \left( {{{\left( {\dfrac{{\log n}}{n}} \right)}^{{\textstyle{1 \over 4}}}}} \right) and d \in \{ 1,2\}, the union of a digraph on n vertices with a minimum degree of at least \alpha n and a random d-regular digraph on n vertices is a.a.s. pancyclic. Moreover, a polynomial-time algorithm is proposed to find cycles of any length in such a digraph.
Graphical Abstract
Minimum degree conditons on Hamilton cycles.
Abstract
Dirac’s theorem states that if a graph G on n vertices has a minimum degree of at least \displaystyle \frac{n}{2}, then G contains a Hamiltonian cycle. Bohman et al. introduced the random perturbed graph model and proved that for any constant \alpha > 0 and a graph H with a minimum degree of at least \alpha n , there exists a constant C depending on α such that for any p \geqslant \dfrac{C}{n}, H \cup {G_{n,p}} is asymptotically almost surely (a.a.s.) Hamiltonian. In this study, the random perturbed digraph model is considered, and we show that for all \alpha = \omega \left( {{{\left( {\dfrac{{\log n}}{n}} \right)}^{{\textstyle{1 \over 4}}}}} \right) and d \in \{ 1,2\}, the union of a digraph on n vertices with a minimum degree of at least \alpha n and a random d-regular digraph on n vertices is a.a.s. pancyclic. Moreover, a polynomial-time algorithm is proposed to find cycles of any length in such a digraph.
Public Summary
If the minimal degree condition is satisfied, then by perturbing the digraph with a random 1-regular graph, the resulting digraph is a.a.s. pancyclic.
Moreover, we give a polynomial algorithm to find cycles of all possible lengths in such perturbed digraph.
Gas metal arc welding (GMAW) heats and melts the metal using an established arc between a continuously fed filler wire electrode and the workpiece metal. It is widely used in marine ships due to its good forming quality, low cost, easy mechanization, and high efficiency[1–3]. The GMAW process involves metal transfer, arc exotherm, and weld pool convection. The complex physical phenomena in the welding process may lead to defects such as undercuts and incomplete fusion in welded joints, which affect the welding quality. The welding quality depends on the morphologies of the weld seams. Weld pool behaviors, microstructures, and the mechanical and metallurgical properties of weld seams are affected by welding parameters such as plate thickness, groove angle, shielding gas flow, arc voltage, and welding speed[4].
Multiple studies have focused on the influence of welding parameters on the macroscopic morphologies and shaping quality of weldments. Sivakumar et al.[5] discussed the effects of process parameters on the cross-sectional area, height, width, and microhardness of weld seams. The results indicate that the welding current is the most significant. Meena et al.[6] discussed the effects of welding current, speed, plate thickness, and gas flow on the geometric features of weld beads through experiments. The voltage significantly affects the weld reinforcement, width, and penetration of welded joints.
Recently, mathematical models and optimization algorithms have been widely used to determine the welding input parameters corresponding to the required welding quality, which establishes the mathematical relationship between the input parameters of the welding process and the output variables of welded joints. Mishra et al.[7] explored the effect of process parameters on the dilution rate of welded joints by response surface methodology (RSM). After that, a mathematical model is established by a second-order regression equation. Ghosh et al.[8] discussed the influence of currents and gas flow on the weld quality of arc welding. The process parameters are optimized by the Gray-Taguchi method to determine the optimal parameter combination. Ramirez et al.[9] analyzed the angular deformations of the GMAW backing weld process at different sizes and thicknesses based on back-propagation neural networks (BPNNs). Furthermore, the accuracy of the model is proven by the verification test.
Consequently, the GMAW process involves complex physical phenomena and multiple factors affecting the quality of welded joints. It is difficult to measure the weld quality by a single evaluation index. Therefore, multiobjective optimizations are necessary for the quality of welded joints. However, multiple-objective optimization problems are usually converted into single-objective optimization in traditional multiobjective optimization[10, 11]. After that, the mathematical optimization method is used to solve the single-objective optimization, which fails to reach the optimal interval for all objectives. In contrast, the nondominated sorting algorithm Ⅱ (NSGA-Ⅱ) can obtain the optimal solution set satisfying all response objectives by the global optimization of input variables.
Therefore, this work explored the effects of different process parameters on the aspect ratio, hardness, and residual stress of weld seams based on mixed-level orthogonal design and regression analysis. Multiobjective optimization by the NSGA-Ⅱderives the optimal process parameter set of minimum aspect ratio, maximum hardness, and minimum residual stress, which provides a theoretical direction for industrial applications.
2.
Experimental equipments and methods
Q355C steel with the characteristics of high strength, toughness, and corrosion resistance is one of the most commonly used materials for offshore wind power towers. The substrate made of Q355C marine steel has dimensions of 100 mm × 60 mm × 10 mm. The diameter of welding wire E71T-1C is 1.2 mm. Table 1 lists the chemical compositions of the Q355C steel and welding wires.
Table
1.
Chemical compositions of Q355C steel and welding wire (mass fraction, %).
The equipment used in this work is the GMAW welding system (see Fig. 1), including the M-10iA industrial robot (FANUC, Japan), INVERTEC CV350-R welding machine (LINCOIN, USA), and AutodriveTM4r90 wire feeder (LINCOIN, USA). CO2 with 99.9% purity is used as the shielding gas during the welding process.
The steel surface was cleaned with sandpapers and acetone before welding to remove contaminants such as rust and grease. A resistance heater was used to preheat the steel plate for more than 5 min to ensure that the steel plate was preheated evenly. Then, the plate was moved to the welding table for welding. Samples were cut, ground, polished, and corroded by wire electric discharge after welding, and the macroscopic metallography of the weld bead was observed. Base metals were completely corroded and then washed with alcohol for drying after 30–40 s of etching by a 4% nitric acid alcohol solution. The hardness of the weldments was measured by a microhardness tester (MVA-402TS, HDNS, China) under a load of 300 g and a dwell time of 10 s at the cross-section of the welded joints. The residual stress of the weld seams was measured by a Canadian Proto X-ray residual stress tester. X-rays emitted by the probe were used to measure residual stress at equal distances from left to right. The average measurement result was the residual stress value of the welding seams. Stress analysis was conducted by the sinφ2 method according to changes in the positions of diffraction peaks in more than two directions.
The test used the Taguchi orthogonal design method to explore the influence of process parameters on the forming quality of welded joints. Table 2 shows the factor levels of the orthogonal test.
Table
2.
Factors and levels of the orthogonal experimental design.
where W is the weld width and H is the weld reinforcement.
The weld width and reinforcement were measured by Digimizer software. The aspect ratio of the weldments was obtained after calculations. The aspect ratio was the main factor affecting the solidification and cracking of structural steel joints. The smaller aspect ratio indicated better weld-metal spreading ability and a smaller tendency of solidification and cracking to a certain extent at the macro level. Table 3 shows the experimental design and results.
Table
3.
Orthogonal experimental design and results.
Experimental data were subjected to regression analysis by the step-back technique. The second-order and interaction terms were used to establish the regression model. Eqs. (2)–(4) show the regression model established by the step-back technique. Table 4 shows the corresponding variance analysis.
Table
4.
Variance analysis for process responses(DF: degree of freedom; Seq. SS: sum of squares of mean deviation; Adj. MS: adjust the mean square; Contribution: the influence of input on output).
The R2 value represents the model data goodness of fit. The higher the R2 value, the higher the goodness of model fitting data. The adjusted R2 value in the work considers the number of independent variables to predict the target variable. It is used to determine whether adding new variables to the model will increase the fitting degree of the model. The predicted R2 value is used to determine the degree to which the model predicts the response of the new observation. The higher the predicted R2value, the better the model’s predictive ability.
Table 4 shows the analysis of variance (ANOVA) for process responses. The p value of the aspect ratio regression model is less than 0.001. R2, adjusted R2, and predicted R2 are all close to 1. The difference between the adjusted R2 and predicted R2 is less than 0.2, indicating that the aspect ratio regression model has high fitting accuracy. The significance analysis shows that the welding voltage, welding current, and their square terms significantly affect the aspect ratio.
According to the ANOVA of hardness and residual stress, the model meets the above requirements. Based on the significance analysis, hardness is significantly affected by the preheating temperature, welding voltage, current, speed, and wire extension. The preheating temperature, welding voltage, and welding speed significantly affect the residual stress. Meanwhile, hardness is significantly affected by the interaction term of preheating temperature and welding current, the interaction term of welding voltage and welding current, the interaction term of welding current and welding speed, welding currents, and wire extension. Residual stress is significantly affected by the square term of the welding voltage and the square of the welding speed term, the interaction term of preheat temperature and welding voltage, and the interaction term of the preheat temperature and welding speed.
3.1
Main effects analysis
Fig. 3a shows the main effect of the aspect ratio model. The aspect ratio decreases with increasing welding voltage, and it first increases and then decreases with increasing welding current because the increased welding voltage amplifies the heat input of the molten pool. The molten state of the molten pool becomes longer, which increases the width and decreases the height[12]. Thus, the aspect ratio decreases. When the welding current appropriately increases, a proportionate increase in molten welding wires leads to a constant weld width and increased height, which slightly enlarges the aspect ratio. Linear energy per unit length of the weld bead amplifies with the increased welding current to increase the volume of the molten substrate and width[13]. Meanwhile, the surface tension decreases with the increased core temperature of the molten pool. Driven by the varying surface tension, the fluidity of molten metal increases, and the spreading ability of the molten pool increases. Thus, the aspect ratio decreases with increasing width.
Figure
3.
Main effects of (a) the aspect ratio model, (b) the hardness model, and (c) the residual stress model.
Fig. 3b presents the main effect of the hardness model. The hardness increases with increasing preheating temperature and welding speed. The hardness decreases with increasing welding voltage, welding current, and wire extension.
Ferrite has the structure of a body-centered cubic lattice, while austenite is made of a face-centered cubic lattice with high interstitial carbon solubility. Martensite forms during the solidification process when carbon atoms cannot diffuse out of austenite. The carbon content in the supercooled austenite significantly decreases when there is enough time for carbon to diffuse out of austenite. Supercooled austenite is transformed into ferrite in this case[14].
The melting rate of welding wires increases with increasing preheating temperature. A reduced melting time shortens the time for carbon to diffuse out of austenite, which promotes the martensitic phase transition. Martensite has a much higher microhardness than ferrite due to the solid solution and phase change strengthening during the martensitic phase transition. The increased preheating temperature promotes the formation of martensite, which increases the hardness. The heat input increases with increasing welding voltage. Excess energy promotes grain growth and coarsening, which decreases hardness.
As the welding current increased, the cooling rate of molten metals slow down which to promote the formation of ferrite. Thus, the hardness decreases. As the welding speed increases, the solidification growth and cooling of weld seams accelerate to produce finer grains[15, 16], which increases hardness. The melting amount of welding wires increases linearly with increasing wire extension. More energy per unit length is required to melt the welding wire. Therefore, the solidification growth rate of the molten metal slows down to decrease the hardness.
Residual stress is affected by the cooling rate, phase changes in the fusion, and heat-affected zones[17]. Tensile residual stress decreases by reducing the temperature gradient at a low heat input[18]. The high heat input and cooling rate increase the tensile residual stress. Fig. 3c presents the main effect of the residual stress model. The residual stress increases with increasing preheating temperature. The molten pool is quickly cooled and heated due to the local concentration and motion of the welding heat source. Part of the energy is used to melt the welding wire to form a molten pool under other unchanged conditions, and part of the heat is transmitted to the environment and the substrate.
When the preheating temperature is too high, the energy transmitted by heat to the substrate decreases, and the maximum temperature of the molten pool increases. When the welding speed is certain, the temperature differences increase, and the residual stress increases. Meanwhile, it first decreases and then increases with increasing welding voltage and speed. The welding current and wire extension do not affect the residual stress. The increased dry elongation of the welding wire linearly increases the melting amount of the welding wire, and the solidification and growth rate of the molten metal slows down. The volume expansion unevenness increases, and the residual stress generally increases.
3.2
Interaction analysis
Fig. 4a shows that the hardness increases with increasing welding current and preheating temperature. The increased welding current increases the heat input, and the increased preheating temperature raises the temperature of the molten pool. The welding current and preheating temperature increase the molten state of the molten pool. The dendrites have sufficient time to extend toward the fusion zone, which reduces the dendrite spacing and size[19]. Therefore, the hardness value increases with decreasing dendrite size[15].
Figure
4.
Surface plots of (a) the hardness, I, and T ; (b) the hardness, U, and I; (c) the hardness, I, and S; (d) the hardness, I, and L.
The welding current and voltage determine the linear energy in the welding process. Fig. 4b shows that when the welding voltage and welding current are small, the hardness is high. The cooling rate of the molten metal increases with decreasing linear energy[20]. A higher cooling rate contributes to the formation of martensite, which increases the hardness. The heat input per unit time increases with increasing welding current and voltage to slow the cooling rate. Excess energy promotes grain growth, which results in a larger dendrite size and a concomitant decrease in hardness[21]. The hardness increases with increasing welding voltage and current. A slower cooling rate (higher heat input) promotes the transformation of unstable ferrites to austenite[22], which decreases the ferrite content and increases the hardness.
Fig. 4c shows that the hardness first decreases and then increases with increasing welding current and speed. The welding current affects the heat input, while the welding speed impacts the heat input, the amount of deposited metal, and the cooling rate. The increased welding current increases the heat input in the welding zone. Higher heat input leads to a slower cooling rate, larger grain size, and decreased hardness. However, the cooling rate increases with increasing welding speed. The solidification rate of the molten metal is increased to refine grains[23], which increases hardness.
Fig. 4d shows that the hardness first increases and then decreases with increasing welding current and wire elongation. The heat input increases, and the melting amount of the weld increases. When the heat input is low, the weldments are cooled at a high rate. Then, grains are refined to increase hardness. The accumulation of welding wires increases with increasing welding current and wire elongation. Meanwhile, high heat input leads to the further growth of grains, which decreases hardness.
Fig. 5a shows that the residual stress first decreases and then increases with increasing preheating temperature and welding voltage. The welding voltage increases to amplify the heat input. Meanwhile, the preheating temperature increases to reduce the temperature gradient, which decreases the residual stress. As the welding voltage increases further, the heat input constantly increases, and the temperature of the molten pool increases. The increased temperature gradient increases the residual stress.
Figure
5.
Surface plots of (a) the residual stress, T, and U; (b) the residual stress, T, and S.
Fig. 5b shows that the residual stress first decreases and then increases with increasing preheating temperature and welding speed. The cooling rate of the molten metal increases with increasing welding speed. When the welding speed is small, accelerated cooling slows the phase transition in the fusion zone, which gradually decreases the length and spacing of dendrites. The temperature gradient of the molten pool decreases at the preheating temperature to reduce residual stress. Accelerated cooling promotes the phase change process in the fusion zone with a further increase in welding speed. The phase transition leads to uneven volume expansions, which increases the residual stress.
4.
Multiobjective optimizations and verification
Achieving maximum hardness, minimum aspect ratio, and residual stress is a typical multiobjective problem. The NSGA-Ⅱ is used for the multiobjective optimization of process parameters in this work. It is a multiobjective evolutionary algorithm based on Pareto solution sets with the elitist strategy and fast nondominated sorting and converges on the Pareto optimal front end. Compared with traditional multiobjective algorithms, the NSGA-Ⅱ does not need to convert multiple objectives into single-objective problems. Multiobjective optimization of the GMAW process parameters is achieved by the NSGA-Ⅱ to obtain the compromise solution among the three optimization objectives.
Fig. 6a shows the flow of the NSGA-Ⅱ. The NSGA-Ⅱ starts with an initialized population Pt that generates offspring Ot through selection, crossover, and mutation operators. Once offspring are obtained, the current population and offspring are combined into one set for sorting according to the nondominated and crowding distances. Finally, a new population Pt+1 can be obtained from the optimal N individuals in the combination set. Table 5 indicates the relevant parameter settings of the NSGA-Ⅱ.
Gamultiobj, is a multi-objective optimization function based on NSGA-Ⅱ, aims at minimizing objective functions. Therefore, we take the opposite of the larger-the-better target value. Fig. 6b shows the Pareto optimal frontier of the aspect ratio, hardness, and residual stress, and each point is a specific optimal solution. Table 6 presents the partially optimized results of the Pareto frontier, and the corresponding process parameters are selected according to actual industrial requirements.
Table
6.
Partial optimized results of the Pareto frontier.
The compromise solutions calculated by the NSGA-Ⅱ are all noninferior solutions. In other words, there are no other solutions that are better than these noninferior solutions in three optimization objectives at the same time. Fig. 6c–d show the smooth Pareto optimal surface and contour plots formed by 70 noninferior solutions. The aspect ratio and residual stress are decreased to increase hardness. Therefore, there is a noninferior solution that optimizes the three objectives simultaneously.
A noninferior solution (see Fig. 6c for the red point) is selected from the Pareto optimal surface to verify the effectiveness of the optimization method. Table 7 indicates the verification parameters and results. Errors between the predicted and experimental values are 9.30%, 5.00%, and 3.07% (all are less than 10%) for the aspect ratio, hardness, and residual stress, respectively. Multiobjective optimization is achieved by the proposed method. In addition, the model established by regression fitting is accurate and reliable. Fig. 7 shows the weld bead profile and size of the verification experiment.
The microstructure of the weld seam was observed by a scanning electron microscope (High-Technologies TM3030Plus, Hitachi, Japan). An attached energy dispersive spectrometer (Model 550i, IXRF, America) was used for compositional analysis, which determined the quality of the optimal weld seam.
Fig. 8a shows the microstructure of the weld seam, including pearlite, proeutectoid ferrite, and acicular ferrite. The acicular ferrite is mainly grown in the prior austenite crystal. The formation of acicular ferrite in the weld is beneficial to improving the strength and toughness of the weld. Ferrite can improve the corrosion resistance of the welded joint to a certain extent and prevent the occurrence of hot cracks in the weld seam. The restriction of the nonequilibrium state in the welding process leads to insufficient diffusion of atoms. The transformation of pearlite is restrained to reduce the pearlite content in the weld zone[24].
Figure
8.
Microstructure morphology and energy spectrum of welded joints (a) weld, (b) heat affected zone, and (c) base metal.
Fig. 8b shows the microstructure of the heat-affected zone, which is mainly composed of granular ferrite, lath ferrite, and granular bainite. The microstructure of the base metal is mainly composed of pearlite and granular ferrite (see Fig. 8c). According to the energy dispersive spectroscopy (EDS) spectrum, the chromium in the weld seam and heat-affected zone is concentrated in acicular ferrite and lath ferrite. The content of alloying elements such as chromium is higher than that in the base metal. The deformed dislocations are hindered in the diffusion to improve the quality of the weld zone due to the grain morphology and solution-induced lattice distortion of these elements. The optimized sample achieves high hardness under a low aspect ratio and residual stress. The segregation of alloy elements appears in the weld seam and heat-affected zone. Therefore, the improved welding quality is better than that of the orthogonal test group.
5.
Conclusions
The orthogonal mixed-level experimental design and fitting regression analysis were used to explore the influence law of GMAW process parameters on the aspect ratio, hardness, and residual stress of weld seams in this work. Multiobjective optimization was conducted by the NSGA-Ⅱ to verify the reliability of the model. The main conclusions are described as follows:
(Ⅰ) The aspect ratio decreased with increasing welding voltage, and it first increased and then decreased with increasing welding current. The hardness increased with increasing preheating temperature and welding speed and decreased with increasing welding voltage, current, and dry extension. Residual stress increased with increasing preheating temperature, and it first decreased and then increased with increasing welding voltage and speed.
(Ⅱ) The relative errors between the predicted and experimental values for the aspect ratio, hardness, and residual stress were all less than 10% under the optimized process conditions. The fitting regression model and NSGA-Ⅱ were combined for the multiobjective optimization of maximum hardness, minimum aspect ratios, and residual stress. The welding parameters were selected from the Pareto optimal frontier according to actual industrial requirements.
Conflict of Interest
The authors declare that they have no conflict of interest.
If the minimal degree condition is satisfied, then by perturbing the digraph with a random 1-regular graph, the resulting digraph is a.a.s. pancyclic.
Moreover, we give a polynomial algorithm to find cycles of all possible lengths in such perturbed digraph.
Dirac G A. Some theorems on abstract graphs. Proceedings of the London Mathematical Society,1952, s3-2 (1): 69–81. DOI: 10.1112/plms/s3-2.1.69
[2]
Korsunov A D. Solution of a problem of Erdös and Rényi on Hamiltonian cycles in undirected graphs. Metody Diskret. Analiz.,1977, 31: 17–56.
[3]
Bohman T, Frieze A, Martin R. How many random edges make a dense graph Hamiltonian? Random Structures Algorithms,2003, 22 (1): 33–42. DOI: 10.1002/rsa.10070
[4]
Krivelevich M, Kwan M, Sudakov B. Cycles and matchings in randomly perturbed digraphs and hypergraphs. Combinatorics, Probability and Computing,2016, 25 (6): 909–927. DOI: 10.1017/S0963548316000079
[5]
Hahn-Klimroth M, Maesaka G S, Mogge Y, et al. Random perturbation of sparse graphs. The Electronic Journal of Combinatorics,2021, 28 (2): P2.26. DOI: 10.37236/9510
[6]
Bedenknecht W, Han J, Kohayakawa Y, et al. Powers of tight Hamilton cycles in randomly perturbed hypergraphs. Random Structures & Algorithms,2019, 55 (4): 795–807. DOI: 10.1002/rsa.20885
[7]
Han J, Zhao Y. Hamiltonicity in randomly perturbed hypergraphs. Journal of Combinatorial Theory, Series B,2020, 144: 14–31. DOI: 10.1016/j.jctb.2019.12.005
Condon P, Espuny Díaz A, Girão A, et al. Hamiltonicity of random subgraphs of the hypercube. In: Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA). Philadelphia, PA: SIAM, 2021.
[10]
Dudek A, Reiher C, Ruciński A, et al. Powers of Hamiltonian cycles in randomly augmented graphs. Random Structures & Algorithms,2020, 56 (1): 122–141. DOI: 10.1002/rsa.20870
[11]
Böttcher J, Montgomery R, Parczyk O, et al. Embedding spanning bounded degree graphs in randomly perturbed graphs. Mathematika,2020, 66 (2): 422–447. DOI: 10.1112/mtk.12005
[12]
Balogh J, Treglown A, Wagner A Z. Tilings in randomly perturbed dense graphs. Combinatorics, Probability and Computing,2019, 28 (2): 159–176. DOI: 10.1017/S0963548318000366
[13]
Han J, Morris P, Treglown A. Tilings in randomly perturbed graphs: Bridging the gap between Hajnal-Szemerédi and Johansson-Kahn-Vu. Random Structures & Algorithms,2021, 58 (3): 480–516. DOI: 10.1002/rsa.20981
Cooper C, Frieze A, Molloy M. Hamilton cycles in random regular digraphs. Combinatorics, Probability and Computing,1994, 3 (1): 39–49. DOI: 10.1017/S096354830000095X
[16]
Bollobás B. A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European Journal of Combinatorics,1980, 1 (4): 311–316. DOI: 10.1016/S0195-6698(80)80030-8
[17]
Wormald N C. Models of random regular graphs. In: Surveys in Combinatorics. Cambridge, UK: Cambridge University Press, 1999.
Dirac G A. Some theorems on abstract graphs. Proceedings of the London Mathematical Society,1952, s3-2 (1): 69–81. DOI: 10.1112/plms/s3-2.1.69
[2]
Korsunov A D. Solution of a problem of Erdös and Rényi on Hamiltonian cycles in undirected graphs. Metody Diskret. Analiz.,1977, 31: 17–56.
[3]
Bohman T, Frieze A, Martin R. How many random edges make a dense graph Hamiltonian? Random Structures Algorithms,2003, 22 (1): 33–42. DOI: 10.1002/rsa.10070
[4]
Krivelevich M, Kwan M, Sudakov B. Cycles and matchings in randomly perturbed digraphs and hypergraphs. Combinatorics, Probability and Computing,2016, 25 (6): 909–927. DOI: 10.1017/S0963548316000079
[5]
Hahn-Klimroth M, Maesaka G S, Mogge Y, et al. Random perturbation of sparse graphs. The Electronic Journal of Combinatorics,2021, 28 (2): P2.26. DOI: 10.37236/9510
[6]
Bedenknecht W, Han J, Kohayakawa Y, et al. Powers of tight Hamilton cycles in randomly perturbed hypergraphs. Random Structures & Algorithms,2019, 55 (4): 795–807. DOI: 10.1002/rsa.20885
[7]
Han J, Zhao Y. Hamiltonicity in randomly perturbed hypergraphs. Journal of Combinatorial Theory, Series B,2020, 144: 14–31. DOI: 10.1016/j.jctb.2019.12.005
Condon P, Espuny Díaz A, Girão A, et al. Hamiltonicity of random subgraphs of the hypercube. In: Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA). Philadelphia, PA: SIAM, 2021.
[10]
Dudek A, Reiher C, Ruciński A, et al. Powers of Hamiltonian cycles in randomly augmented graphs. Random Structures & Algorithms,2020, 56 (1): 122–141. DOI: 10.1002/rsa.20870
[11]
Böttcher J, Montgomery R, Parczyk O, et al. Embedding spanning bounded degree graphs in randomly perturbed graphs. Mathematika,2020, 66 (2): 422–447. DOI: 10.1112/mtk.12005
[12]
Balogh J, Treglown A, Wagner A Z. Tilings in randomly perturbed dense graphs. Combinatorics, Probability and Computing,2019, 28 (2): 159–176. DOI: 10.1017/S0963548318000366
[13]
Han J, Morris P, Treglown A. Tilings in randomly perturbed graphs: Bridging the gap between Hajnal-Szemerédi and Johansson-Kahn-Vu. Random Structures & Algorithms,2021, 58 (3): 480–516. DOI: 10.1002/rsa.20981
Cooper C, Frieze A, Molloy M. Hamilton cycles in random regular digraphs. Combinatorics, Probability and Computing,1994, 3 (1): 39–49. DOI: 10.1017/S096354830000095X
[16]
Bollobás B. A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European Journal of Combinatorics,1980, 1 (4): 311–316. DOI: 10.1016/S0195-6698(80)80030-8
[17]
Wormald N C. Models of random regular graphs. In: Surveys in Combinatorics. Cambridge, UK: Cambridge University Press, 1999.
Table
4.
Variance analysis for process responses(DF: degree of freedom; Seq. SS: sum of squares of mean deviation; Adj. MS: adjust the mean square; Contribution: the influence of input on output).
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