76 research outputs found
Damage mechanisms of matrix cracking and interfacial debonding in random fiber composites under dynamic loadings
No full textBy considering the wide applications of composite materials, it is necessary to have a proper knowledge of dynamic behavior as well as static behavior reflecting the damage in composite materials. Strain rates have significant effects on dynamic behavior in composite materials when they are under dynamic loadings. In this thesis, a multiscale numerical approach with finite element code ABAQUS is developed to characterize failure criteria to express static and dynamic damage mechanisms of matrix cracking and interfacial debonding under uniaxial tensile loadings for composite materials. The random epoxy/glass composite material is investigated under three strain rates: quasi-static, intermediate and high, corresponding to 10-4, 1 and 200 s-1, respectively. A representative volume element (RVE) of a random glass fiber composite is employed to analyze microscale damage mechanisms of matrix cracking and interfacial debonding, while the associated damage variables are defined and applied in a mesoscale stiffness reduction law. The macroscopic response of the homogenized damage model is investigated using finite element analysis and validated through experiments. The random epoxy/glass composite specimens fail at a smaller strain; there is less matrix cracking but more interfacial debonding accumulated as the strain rate increases. The dynamic simulation results of stress strain response are compared with experimental tests carried out on composite specimens, and a respectable agreement between them under the low strain rate is observed. Finally, a case study of a random glass fiber composite plate containing a central hole subjected to tensile loading is performed to illustrate the applicability of the multiscale damage model.Ph. D.Includes bibliographical referencesby Wensong Yan
Multicoloring and Mycielski construction
No full textAbstractThe generalized Mycielskians of graphs (also known as cones over graphs) are the natural generalization of the Mycielskians of graphs (which were first introduced by Mycielski in 1955). Given a graph G and any integer p⩾0, one can transform G into a new graph μp(G), the p-Mycielskian of G. In this paper, we study the kth chromatic numbers χk of Mycielskians and generalized Mycielskians of graphs. We show that χk(G)+1⩽χk(μ(G))⩽χk(G)+k, where both upper and lower bounds are attainable. We then investigate the kth chromatic number of Mycielskians of cycles and determine the kth chromatic number of p-Mycielskian of a complete graph Kn for any integers k⩾1, p⩾0 and n⩾2. Finally, we prove that if a graph G is a/b-colorable then the p-Mycielskian of G, μp(G), is (at+bp+1)/bt-colorable, where t=∑i=0p(a-b)ibp-i. And thus obtain graphs G with m(G) grows exponentially with the order of G, where m(G) is the minimal denominator of a a/b-coloring of G with χf(G)=a/b
Strong chromatic index of claw-free graphs with edge weight seven
No full textLet be a graph and a positive integer. A strong -edge-coloring of is a mapping such that for any two edges and e^' that are either adjacent to each other or adjacent to a common edge, \phi(e)\ne \phi(e^'). The strong chromatic index of is the minimum integer such that has a strong -edge-coloring. The edge weight of is defined to be , where denotes the degree of in . In this paper, we prove that every claw-free graph with edge weight at most has strong chromatic index at most , which is sharp
Some star extremal circulant graphs
No full textAbstractThe circular chromatic number χc(G) and the fractional chromatic number χf(G) are two generalizations of the ordinary chromatic number of a graph G. A graph is called star extremal if its circular chromatic number equals its fractional chromatic number. Gao and Zhu (Discrete Math. 152 (1996) 147–156), Lih et al. (SIAM J. Discrete Math. 12 (1999) 491–499) gave many classes of circulant graphs which are star extremal. In this paper, we study the star extremality of circulant graphs whose generating sets are of the form {1,2,…,m−1,k,k+1,…,k+m−2}, {k,k+1,…,k′}, and {k,k+1,…,k1,k2,k2+1,…,⌊p/2⌋}, where p is the vertex number of the graph. As a corollary, we give an improvement of a result of Gao and Zhu (Discrete Math. 152 (1996) 147–156)
Proving a conjecture on the upper bound of semistrong chromatic indices of graphs
No full textLet be a graph with maximum degree . For a subset
of , we denote by the subgraph of induced by the
endvertices of edges in . We call a semistrong matching if each edge of
is incident with a vertex that is of degree 1 in . Given a
positive integer , a semistrong -edge-coloring of is an edge coloring
using at most colors in which each color class is a semistrong matching of
. The semistrong chromatic index of , denoted by , is the
minimum integer such that has a semistrong -edge-coloring. Recently,
Lu\v{z}ar, Mockov\v{c}iakov\'a and Sot\'ak conjectured that for any connected graph except the complete bipartite graph
. In this paper, we settle this conjecture by proving that
each such graph other than a cycle on vertices has a semistrong edge
coloring using at most colors.Comment: 20 pages, 9 figure
Maximum weight
No full textLet t be a nonnegative integer and G = (V(G),E(G)) be a graph. For v ∈ V(G), let NG(v) = {u ∈ V(G) \ {v} : uv ∈ E(G)}. And for S ⊆ V(G), we define dS(G; v) = |NG(v) ∩ S| for v ∈ S and dS(G; v) = −1 for v ∈ V(G) \ S. A subset S ⊆ V(G) is called a t-sparse set of G if the maximum degree of the induced subgraph G[S] does not exceed t. In particular, a 0-sparse set is precisely an independent set. A vector-weighted graph is a graph G with a vector weight function , where for each v ∈ V(G). The weight of a t-sparse set S in is defined as . And a t-sparse set S is a maximum weight t-sparse set of if there is no t-sparse set of larger weight in . In this paper, we propose the maximum weight t-sparse set problem on vector-weighted graphs, which is to find a maximum weight t-sparse set of . We design a dynamic programming algorithm to find a maximum weight t-sparse set of an outerplane graph which takes O((t + 2)4n) time, where n = |V(G)|. Moreover, we give a polynomial-time algorithm for this problem on graphs with bounded treewidth
Circular game chromatic number of graphs
No full textAbstractIn a circular r-colouring game on G, Alice and Bob take turns colouring the vertices of G with colours from the circle S(r) of perimeter r. Colours assigned to adjacent vertices need to have distance at least 1 in S(r). Alice wins the game if all vertices are coloured, and Bob wins the game if some uncoloured vertices have no legal colour. The circular game chromatic number χcg(G) of G is the infimum of those real numbers r for which Alice has a winning strategy in the circular r-colouring game on G. This paper proves that for any graph G, χcg(G)≤2colg(G)−2, where colg(G) is the game colouring number of G. This upper bound is shown to be sharp for forests. It is also shown that for any graph G, χcg(G)≤2χa(G)(χa(G)+1), where χa(G) is the acyclic chromatic number of G. We also determine the exact value of the circular game chromatic number of some special graphs, including complete graphs, paths, and cycles
On n-fold L(j,k)-and circular L(j,k)-labelings of graphs
No full textAbstractWe initiate research on the multiple distance 2 labeling of graphs in this paper.Let n,j,k be positive integers. An n-fold L(j,k)-labeling of a graph G is an assignment f of sets of nonnegative integers of order n to the vertices of G such that, for any two vertices u,v and any two integers a∈f(u), b∈f(v), |a−b|≥j if uv∈E(G), and |a−b|≥k if u and v are distance 2 apart. The span of f is the absolute difference between the maximum and minimum integers used by f. The n-fold L(j,k)-labeling number of G is the minimum span over all n-fold L(j,k)-labelings of G.Let n,j,k and m be positive integers. An n-fold circular m-L(j,k)-labeling of a graph G is an assignment f of subsets of {0,1,…,m−1} of order n to the vertices of G such that, for any two vertices u,v and any two integers a∈f(u), b∈f(v), min{|a−b|,m−|a−b|}≥j if uv∈E(G), and min{|a−b|,m−|a−b|}≥k if u and v are distance 2 apart. The minimum m such that G has an n-fold circular m-L(j,k)-labeling is called the n-fold circular L(j,k)-labeling number of G.We investigate the basic properties of n-fold L(j,k)-labelings and circular L(j,k)-labelings of graphs. The n-fold circular L(j,k)-labeling numbers of trees, and the hexagonal and p-dimensional square lattices are determined. The upper and lower bounds for the n-fold L(j,k)-labeling numbers of trees are obtained. In most cases, these bounds are attainable. In particular, when k=1 both the lower and the upper bounds are sharp. In many cases, the n-fold L(j,k)-labeling numbers of the hexagonal and p-dimensional square lattices are determined. In other cases, upper and lower bounds are provided. In particular, we obtain the exact values of the n-fold L(j,1)-labeling numbers of the hexagonal and p-dimensional square lattices
The strong chromatic index of a class of graphs
No full textAbstractThe strong chromatic index of a graph G is the minimum integer k such that the edge set of G can be partitioned into k induced matchings. Faudree et al. [R.J. Faudree, R.H. Schelp, A. Gyárfás, Zs. Tuza, The strong chromatic index of graphs, Ars Combin. 29B (1990) 205–211] proposed an open problem: If G is bipartite and if for each edge xy∈E(G), d(x)+d(y)≤5, then sχ′(G)≤6. Let H0 be the graph obtained from a 5-cycle by adding a new vertex and joining it to two nonadjacent vertices of the 5-cycle. In this paper, we show that if G (not necessarily bipartite) is not isomorphic to H0 and d(x)+d(y)≤5 for any edge xy of G then sχ′(G)≤6. The proof of the result implies a linear time algorithm to produce a strong edge coloring using at most 6 colors for such graphs
On t-relaxed chromatic number of r-power paths
No full text Let [Formula: see text] be a graph and [Formula: see text] a non-negative integer. Suppose [Formula: see text] is a mapping from the vertex set of [Formula: see text] to [Formula: see text]. If, for any vertex [Formula: see text] of [Formula: see text], the number of neighbors [Formula: see text] of [Formula: see text] with [Formula: see text] is less than or equal to [Formula: see text], then [Formula: see text] is called a [Formula: see text]-relaxed [Formula: see text]-coloring of [Formula: see text]. And [Formula: see text] is said to be [Formula: see text]-colorable. The [Formula: see text]-relaxed chromatic number of [Formula: see text], denote by [Formula: see text], is defined as the minimum integer [Formula: see text] such that [Formula: see text] is [Formula: see text]-colorable. Let [Formula: see text] and [Formula: see text] be two positive integers with [Formula: see text]. Denote by [Formula: see text] the path on [Formula: see text] vertices and by [Formula: see text] the [Formula: see text]th power of [Formula: see text]. This paper determines the [Formula: see text]-relaxed chromatic number of [Formula: see text] the [Formula: see text]th power of [Formula: see text]. </jats:p
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