Characterization Results for the L(2, 1, 1)-Labeling Problem on Trees

An L(2, 1, 1)-labeling of a graph G is an assignment of non-negative integers (labels) to the vertices of G such that adjacent vertices receive labels with difference at least 2, and vertices at distance 2 or 3 receive distinct labels. The span of such a labelling is the difference between the maximum and minimum labels used, and the minimum span over all L(2, 1, 1)-labelings of G is called the L(2, 1, 1)-labeling number of G, denoted by λ2,1,1(G). It was shown by King, Ras and Zhou in [The L(h, 1, 1)-labelling problem for trees, European J. Combin. 31 (2010) 1295–1306] that every tree T has Δ2(T) − 1 ≤ λ2,1,1(T) ≤ Δ2(T), where Δ2(T) = maxuv∈E(T)(d(u) + d(v)). And they conjectured that almost all trees have the L(2, 1, 1)-labeling number attain the lower bound. This paper provides some sufficient conditions for λ2,1,1(T) = Δ2(T). Furthermore, we show that the sufficient conditions we provide are also necessary for trees with diameter at most 6

Recent progress on strong edge-coloring of graphs

A strong edge-coloring of a graph G = (V,E) is a partition of its edge set E into induced matchings. In this paper, we gave a short survey on recent results about strong edge-coloring of a graph

On Critical Circuits in k-Connected Matroids

We show that, for every integer k≥ 4 , if M is a k-connected matroid and C is a circuit of M such that for every e∈ C, M\ e is not k-connected, then C meets a cocircuit of size at most 2 k- 3 ; furthermore, if M is binary and k≥ 5 , then C meets a cocircuit of size at most 2 k- 4. It follows from our results and a result of Reid et al [5] that every minimally k-connected matroid has a cocircuit of size at most 2 k- 3 , and every minimally k-connected binary matroid has a cocircuit of size at most 2 k- 4

On Critical Circuits in k-Connected Matroids

We show that, for every integer k≥ 4 , if M is a k-connected matroid and C is a circuit of M such that for every e∈ C, M\ e is not k-connected, then C meets a cocircuit of size at most 2 k- 3 ; furthermore, if M is binary and k≥ 5 , then C meets a cocircuit of size at most 2 k- 4. It follows from our results and a result of Reid et al [5] that every minimally k-connected matroid has a cocircuit of size at most 2 k- 3 , and every minimally k-connected binary matroid has a cocircuit of size at most 2 k- 4

Strong edge-coloring for planar graphs with large girth

A strong edge-coloring of a graph [Formula presented] is a partition of its edge set [Formula presented] into induced matchings. Let [Formula presented] be a connected planar graph with girth [Formula presented] and maximum degree [Formula presented]. We show that either [Formula presented] is isomorphic to a subgraph of a very special [Formula presented]-regular graph with girth [Formula presented], or [Formula presented] has a strong edge-coloring using at most [Formula presented] colors

Enumerating stereo-isomers of tree-like polyinositols

NSFC [11271307]Enumeration of molecules is one of the fundamental problems in bioinformatics and chemoinformatics which is also important from a practical viewpoint. We consider the problem of enumerating the stereo-isomers of tree-like polyinositol molecules (with chemical formula where is the number of hexagonal oinositol rings) and monosubstituted tree-like polyinositols (with chemical formula ). We establish recursion counting formulas for the numbers of the stereo-isomers for these two classes of molecules, in which chirality is also taken into account. In our study, the generating function, Plya enumeration theory and 'Dissimilarity Characteristic Theorem' play important roles. Compared to some known computer programs such as ISOMERS, MOLGEN, exhaustive construction and Dynamic Programming etc., our method is more efficient to our enumeration problem with larger number of inositol rings. Further more, based on the obtained recursion formulas, we derive the asymptotic values for the numbers of these two stereo-isomers from which we conclude that almost all tree-like and monosubstituted tree-like polyinositols are chiral

DETERMINING THE COMPONENT NUMBER OF LINKS CORRESPONDING TO TRIANGULAR AND HONEYCOMB LATTICES

NSFC [10831001]; Fundamental Research Funds for the Central Universities [2010121007]There is a classical correspondence between edge-signed plane graphs and link diagrams. Determining component number of links corresponding to plane graphs may be one of the first problems in studying links by using graphs. There has been several early studies in this aspect, for example, the component number of links formed from 2-dimensional square lattices (4(4)) has been determined. In this paper, we determine the component number of links corresponding to 2-dimensional triangular (3(6)) and honeycomb (6(3)) lattices with free or cyclic boundary condition

Enumerating the total colorings of a polyhedron and application to polyhedral links

In this paper we consider the enumeration problem of a particular three-dimensional molecular or chemical compound system which has a polyhedral frame where the vertices, edges and faces represent 'units' such as atoms, bonds, ligands, polymers, or other objects of chemical interests. In this system, chirality is also taken into account. This enumeration problem is mathematically modeled as the 'total coloring' enumeration problem of a polyhedron: i.e., the number of ways to color all the vertices, edges and faces of the polyhedron by using three or more corresponding color sets, in which some colors may be chiral. We establish a general formula for this enumeration problem by extending the fundamental version of Plya's enumeration theorem. In particular, we apply this technique to the enumeration problem of polyhedral links which have received special attention from biochemists, mathematical chemists and mathematicians over the past two decades.NSFC [10831001

Enumerating DNA polyhedral links

NSFC [10831001]With the fast-growing of DNA self-assembly techniques, a large number of DNA catenanes or more specifically, DNA polyhedral links were synthesized in the past decade. As a sequel of a recent paper (J. Math. Chem. 50, p. 1693, 2012) by the present authors on the enumeration problem of polyhedral links, this paper considers the enumeration problem of DNA polyhedral links. In contrast to a general molecular link, a DNA polyhedral link has four notable features: 1. the memory of DNA chain direction; 2. the accurate DNA complementary base pairing; 3. the twist patterns of the double-helical strands; 4. the migration in branched junction. These features put forward particular requests for treating the enumeration problem of DNA polyhedral links. In addition to using the standard Plya's counting theory, we here introduce the generating function and edge direction retentivity analysis to the enumeration problem, by which we establish explicit expressions of the numbers of DNA polyhedral links for three typical models. These models have been used as strategies or are potential strategies in the synthesis of DNA polyhedral links