Numerical analysis for a system coupling curve evolution attached orthogonally to a fixed boundary,to a reaction–diffusion equation on the curve

We consider a semidiscrete finite element approximation for a system consisting of the evolution of a planar curve evolving by forced curve shortening flow inside a given bounded domain , such that the curve meets the boundary orthogonally, and the forcing is a function of the solution of a reaction–diffusion equation that holds on the evolving curve. We prove optimal order error bounds for the resulting approximation and present numerical experiments.     

H1-superconvergence of finite difference method based on Q1-element on quasi-uniform mesh for the 3D Poisson equation

In this paper, the finite difference (FD) method is considered for the 3D Poisson equation by using the Q₁-element on a quasi-uniform mesh. First, under the regularity assumption of , the H¹-superconvergence of the FD solution u_h based on the Q₁-element to the first-order interpolation function is obtained. Next, the H¹-superconvergence of the second-order interpolation postprocessing function based on the FD solution u_h to u is provided. Finally, numerical tests are presented to show the H¹-superconvergence result of the FD postprocessing function to u if .     

A robust scheme based on novel-operational matrices for some classes of time-fractional nonlinear problems arising in mechanics and mathematical physics

In this paper, we present a novel approach based on shifted Gegenbauer wavelets to attain approximate solutions of some classed of time-fractional nonlinear problems. First, we present the approximation of a function of two variables u(x,t) with help of shifted Gegenbauer wavelets and then some novel operational matrices are proposed with the help of piecewise functions to investigate the positive integer derivative ( D _x and D _t), fractional-order derivative ( and ), fractional-order integration ( and ) and delay terms ( and ) of approximated function u(x,t). In order to transform the discussed nonlinear problem into linear problem Picard iterative scheme has been adopt. The current scheme converts the discussed highly nonlinear time-fractional problem into system of linear algebraic equation the help of developed operational matrices and Picard idea. Analysis on the error bound and convergence to authenticate the mathematical formulation of the computational algorithm. We solve various test problems, such as the van der Pol oscillator model, generalized Burger–Huxley, neutral delay parabolic differential equations, sine-Gordon, parabolic integro-differential equation and nonlinear Schrödinger equations to show the efficiency and accuracy of linearized shifted Gegenbauer wavelets method. A comprehensive comparative examination shows the credibility, accuracy, and reliability of the presently proposed computational approach. Also, this scheme can be extended conveniently to other multi-dimensional physical problems of highly nonlinear fractional or variable order of complex nature.     

Superconvergence analysis of an energy stable scheme with three step backward differential formula-finite element method for nonlinear reaction–diffusion equation

A three step backward differential formula scheme is proposed for nonlinear reaction–diffusion equation and superconvergence results are studied with Galerkin finite element method unconditionally. Energy stability is testified for the constructed scheme with an artificial term. Splitting technique is utilized to get rid of the ratio between the time step size and the subdivision parameter . Temporal error estimate in H²-norm is derived, which leads to the boundedness of the solutions of the time-discrete equations. Unconditional spatial error estimate in L²-norm is deduced which help bound the numerical solutions in L^∞-norm. Superconvergent property of in H¹-norm with order is obtained by taking difference between two time levels of the error equations unconditionally. The global superconvergent property is deduced through the above results. Two numerical examples show the validity of the theoretical analysis.     

On the numerical approximation of p‐biharmonic and ∞‐biharmonic functions

The ∞ ‐Bilaplacian is a third‐order fully nonlinear PDE given by (1) In this work, we build a numerical method aimed at quantifying the nature of solutions to this problem, which we call ∞ ‐biharmonic functions. For fixed p we design a mixed finite element scheme for the prelimiting equation, the p‐Bilaplacian (2) We prove convergence of the numerical solution to the weak solution of and show that we are able to pass to the limit p → ∞ . We perform various tests aimed at understanding the nature of solutions of and we prove convergence of our discretization to an appropriate weak solution concept of this problem that of ‐solutions.     

Extending edge-colorings of complete hypergraphs into regular colorings

Let be the collection of all -subsets of an -set . Given a coloring (partition) of a set , we are interested in finding conditions under which this coloring is extendible to a coloring of so that the number of times each element of appears in each color class (all sets of the same color) is the same number . The case was studied by Sylvester in the 18th century and remained open until the 1970s. The case is extensively studied in the literature and is closely related to completing partial symmetric Latin squares. For , we settle the cases , and completely. Moreover, we make partial progress toward solving the case where . These results can be seen as extensions of the famous Baranyai’s theorem, and make progress toward settling a 40-year-old problem posed by Cameron.     

Minimum codegree condition for perfect matchings in k-partite k-graphs

Let be a -partite -graph with vertices in each partition class, and let denote the minimum codegree of . We characterize those with and with no perfect matching. As a consequence, we give an affirmative answer to the following question of Rödl and Ruciński: if is even or , does imply that has a perfect matching? We also give an example indicating that it is not sufficient to impose this degree bound on only two types of -sets.     

Long properly coloured cycles in edge-coloured graphs

Let be an edge-coloured graph. The minimum colour degree of is the largest integer such that, for every vertex , there are at least distinct colours on edges incident to . We say that is properly coloured if no two adjacent edges have the same colour. In this paper, we show that, for any and large, every edge-coloured graph with contains a properly coloured cycle of length at least .     

Upper bounds on the uniquely restricted chromatic index

Golumbic, Hirst, and Lewenstein define a matching in a simple, finite, and undirected graph to be uniquely restricted if no other matching covers exactly the same set of vertices. We consider uniquely restricted edge-colorings of , defined as partitions of its edge set into uniquely restricted matchings, and study the uniquely restricted chromatic index of , defined as the minimum number of uniquely restricted matchings required for such a partition. For every graph , where is the classical chromatic index, the acyclic chromatic index, and the strong chromatic index of . While Vizing's famous theorem states that is either the maximum degree of or , two famous open conjectures due to Alon, Sudakov, and Zaks, and to Erdős and Nešetřil concern upper bounds on and in terms of . Since is sandwiched between these two parameters, studying upper bounds in terms of is a natural problem. We show that with equality if and only if some component of is . If is connected, bipartite, and distinct from , and is at least , then, adapting Lovász's elegant proof of Brooks’ theorem, we show that . Our proofs are constructive and yield efficient algorithms to determine the corresponding edge-colorings.     

On perfect matchings in matching covered graphs

A graph is matching-covered if every edge of is contained in a perfect matching. A matching-covered graph is strongly coverable if, for any edge of , the subgraph is still matching-covered. An edge subset of a matching-covered graph is feasible if there exist two perfect matchings and such that , and an edge subset with at least two edges is an equivalent set if a perfect matching of contains either all edges in or none of them. A strongly matchable graph does not have an equivalent set, and any two independent edges of form a feasible set. In this paper, we show that for every integer , there exist infinitely many -regular graphs of class 1 with an arbitrarily large equivalent set that is not switching-equivalent to either or , which provides a negative answer to a problem of Lukot’ka and Rollová. For a matching-covered bipartite graph , we show that has an equivalent set if and only if it has a 2-edge-cut that separates into two balanced subgraphs, and is strongly coverable if and only if every edge-cut separating into two balanced subgraphs and satisfies and .     

An improvement on the maximum number of -dominating independent sets

Erdős and Moser raised the question of determining the maximum number of maximal cliques or, equivalently, the maximum number of maximal independent sets in a graph on vertices. Since then there has been a lot of research along these lines. A -dominating independent set is an independent set such that every vertex not contained in has at least neighbors in . Let denote the maximum number of -dominating independent sets in a graph on vertices, and let . Nagy initiated the study of . In this study, we disprove a conjecture of Nagy using a graph product construction and prove that for any even we have We also prove that for any we have improving the upper bound of Nagy.     

Filling the complexity gaps for colouring planar and bounded degree graphs

A colouring of a graph is a function such that for every . A -regular list assignment of is a function with domain such that for every , is a subset of of size . A colouring of respects a -regular list assignment of if for every . A graph is -choosable if for every -regular list assignment of , there exists a colouring of that respects . We may also ask if for a given -regular list assignment of a given graph , there exists a colouring of that respects . This yields the -Regular List Colouring problem. For , we determine a family of classes of planar graphs, such that either -Regular List Colouring is -complete for instances with , or every is -choosable. By using known examples of non--choosable and non--choosable graphs, this enables us to classify the complexity of -Regular List Colouring restricted to planar graphs, planar bipartite graphs, planar triangle-free graphs, and planar graphs with no -cycles and no -cycles. We also classify the complexity of -Regular List Colouring and a number of related colouring problems for graphs with bounded maximum degree.     

On dihedral flows in embedded graphs

Let be a multigraph with for each vertex a cyclic order of the edges incident with it. For , let be the dihedral group of order . Define . Goodall et al in 2016 asked whether admits a nowhere-identity -flow if and only if it admits a nowhere-identity -flow with (a “nowhere-identity dihedral -flow”). We give counterexamples to this statement and provide general obstructions. Furthermore, the complexity of deciding the existence of nowhere-identity -flows is discussed. Lastly, graphs in which the equivalence of the existence of flows as above is true are described. We focus particularly on cubic graphs.     

The size-Ramsey number of powers of paths

Given graphs and and a positive integer , say that is -Ramsey for , denoted , if every -coloring of the edges of contains a monochromatic copy of . The size-Ramsey number of a graph is defined to be . Answering a question of Conlon, we prove that, for every fixed , we have , where is the th power of the -vertex path (ie, the graph with vertex set and all edges such that the distance between and in is at most ). Our proof is probabilistic, but can also be made constructive.     

A dichotomy for the kernel by H-walks problem in digraphs

Let be a digraph which may contain loops, and let be a loopless digraph with a coloring of its arcs . An -walk of is a walk of such that is an arc of , for every . For , we say that reaches by -walks if there exists an -walk from to in . A subset is a kernel by -walks of if every vertex in reaches by -walks some vertex in , and no vertex in can reach another vertex in by -walks. A panchromatic pattern is a digraph such that every -arc-colored digraph has a kernel by -walks. In this study, we prove that every digraph is either a panchromatic pattern, or the problem of determining whether an arc-colored digraph has a kernel by -walks is -complete.     

Linearly χ-bounding (P6, C4)-free graphs*

Given two graphs and , a graph is -free if it contains no induced subgraph isomorphic to or . Let and be the path on vertices and the cycle on vertices, respectively. In this paper we show that for any -free graph it holds that , where and are the chromatic number and clique number of , respectively. Our bound is attained by several graphs, for instance, the 5-cycle, the Petersen graph, the Petersen graph with an additional universal vertex, and all -critical -free graphs other than (see Hell and Huang [Discrete Appl. Math. 216 (2017), pp. 211–232]). The new result unifies previously known results on the existence of linear -binding functions for several graph classes. Our proof is based on a novel structure theorem on -free graphs that do not contain clique cutsets. Using this structure theorem we also design a polynomial time -approximation algorithm for coloring -free graphs. Our algorithm computes a coloring with colors for any -free graph in time.     

Bounded diameter arboricity

We introduce the notion of bounded diameter arboricity. Specifically, the diameter- arboricity of a graph is the minimum number such that the edges of the graph can be partitioned into forests each of whose components has diameter at most . A class of graphs has bounded diameter arboricity if there exists a natural number such that every graph in the class has diameter- arboricity at most . We conjecture that the class of graphs with arboricity at most has bounded diameter arboricity at most . We prove this conjecture for by proving the stronger assertion that the union of a forest and a star forest can be partitioned into two forests of diameter at most 18. We use these results to characterize the bounded diameter arboricity for planar graphs of girth at least for all . As an application we show that every 6-edge-connected planar (multi)graph contains two edge-disjoint -thin spanning trees. Moreover, we answer a question by Mütze and Peter, leading to an improved lower bound on the density of globally sparse Ramsey graphs.     

Reconstruction from the deck of -vertex induced subgraphs

The - deck of a graph is its multiset of subgraphs induced by vertices; we study what can be deduced about a graph from its -deck. We strengthen a result of Manvel by proving for that when is large enough ( suffices), the -deck determines whether an -vertex graph is connected ( suffices when , and cannot suffice). The reconstructibility of a graph with vertices is the largest such that is determined by its -deck. We generalize a result of Bollobás by showing for almost all graphs. As an upper bound on , we have . More generally, we compute whenever , which involves extending a result of Stanley. Finally, we show that a complete -partite graph is reconstructible from its -deck.     

Strong chromatic index of unit distance graphs

The strong chromatic index of a graph , denoted by , is defined as the least number of colors in a coloring of edges of , such that each color class is an induced matching (or: if edges and have the same color, then both vertices of are not adjacent to any vertex of ). A graph is a unit distance graph in if vertices of can be uniquely identified with points in , so that is an edge of if and only if the Euclidean distance between the points identified with and is 1. We would like to find the largest possible value of , where is a unit distance graph (in and ) of maximum degree . We show that , where is a unit distance graph in of maximum degree . We also show that the maximum possible size of a strong clique in unit distance graph in is linear in and give a tighter result for unit distance graphs in the plane.     

Fractal graphs

The lexicographic sum of graphs is defined as follows. Let be a graph. With each associate a graph . The lexicographic sum of the graphs over is obtained from by substituting each by . Given distinct , we have all the possible edges in the lexicographic sum between and if , and none otherwise. When all the graphs are isomorphic to some graph , the lexicographic sum of the graphs over is called the lexicographic product of by and is denoted by . We say that a graph is fractal if there exists a graph , with at least two vertices, such that . There is a simple way to construct fractal graphs. Let be a graph with at least two vertices. The graph is defined on the set of functions from to as follows. Given distinct is an edge of if is an edge of , where is the smallest integer such that . The graph is fractal because . We prove that a fractal graph is isomorphic to a lexicographic sum over an induced subgraph of , which is itself fractal.