Colouring series-parallel graphs

We establish a minimax formula for the chromatic index of series-parallel graphs; and also prove the correctness of a greedy algorithm for finding a vertex-colouring of a series-parallel graph.     

Boxicity of series-parallel graphs

We show that there exist series-parallel graphs with boxicity 3.     

系列平行图SHMeredith图的关联着色

图的关联着色是从关联集到颜色集的一个映射,使得关联集中任何两个相邻的关联都具有不同的像.确定了Meredith图的关联色数,证明了对任意系列平行图都存在一个(Δ 2,2)-关联着色.     

List total colorings of series-parallel graphs

A total coloring of a graph G is a coloring of all elements of G, i.e., vertices and edges, in such a way that no two adjacent or incident elements receive the same color. Let L(x) be a set of colors assigned to each element x of G. Then a list total coloring of G is a total coloring such that each element x receives a color contained in L(x). The list total coloring problem asks whether G has a list total coloring. In this paper, we first show that the list total coloring problem is NP-complete even for series-parallel graphs. We then give a sufficient condition for a series-parallel graph to have a list total coloring, that is, we prove a theorem that any series-parallel graph G has a list total coloring if |L(v)|min{5,Δ+1} for each vertex v and |L(e)|max{5,d(v)+1,d(w)+1} for each edge e=vw, where Δ is the maximum degree of G and d(v) and d(w) are the degrees of the ends v and w of e, respectively. The theorem implies that any series-parallel graph G has a total coloring with Δ+1 colors if Δ4. We finally present a linear-time algorithm to find a list total coloring of a given series-parallel graph G if G satisfies the sufficient condition.     

Quadratic assignment problems on series-parallel digraphs

Christofides has shown that the quadratic assignment problem (QAP) on isomorphic trees is solvable in polynomial time by dynamic programming. We generalize the approach to minimal series-parallel digraphs and show that this problem is equivalent to the general QAP, thus NP-hard. However the restriction to the subclass of series-parallel digraphs not containing bipartite subgraphs again is solvable in polynomial time. An algorithm for this class is given.This work was financially supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung, Projekt S32/01.     

Combinatorial invariants on planar graphs

This paper introduces three kinds of operators on planar graphs with binary weights on edges, for which combinatorial invariants on two kinds of equivalences are found. Further, it is shown that the Jones polynomial and the bracket polynomial which are proved to be new topological invariants on knots in topology become special cases. Moreover, these invariants are a kind of generalization of Tutte polynomial on graphs.Supported by the NNSF of China.     

A compact labeling scheme for series-parallel graphs

A linear time labeling algorithm is presented for series-parallel graphs. The labels enable us to efficiently implement dynamic programming algorithms for sequencing problems with series-parallel precedence constraints. The labeling scheme can also be used to efficiently count and generate the initial sets, terminal sets and independent sets in transitive series-parallel digraphs and to provide a characterization of the maximal independent sets in transitive digraphs.     

On the two-dimensional orthogonal drawing of series-parallel graphs

It has been known that every planar 4-graph has a 2-bend 2-D orthogonal drawing, with the only exception being the octahedron, every planar 3-graph has a 1-bend 2-D orthogonal drawing with the only exception being K₄, and every outerplanar 3-graph with no triangles has a 0-bend 2-D orthogonal drawing. We show in this paper that every series-parallel 4-graph has a 1-bend 2-D orthogonal drawing.     

Combinatorial PDEs on Cayley and coset graphs

Building on the work by E. Barletta and S. Dragomir (2002) [3], this paper solves the initial value problems for the combinatorial heat and wave equations on Cayley and coset graphs.     

Combinatorial aspects of geometric graphs

As a special case of our main result, we show that for all L> 0, each k-nearest neighbor graph in d dimensions excludes K_h as a depth L minor if h = Ω(L^d). More generally, we prove that the overlap graphs defined by Miller, Teng, Thurston and Vavasis (1993) have this combinatorial property. By a construction of Plotkin, Rao and Smith (1994), our result implies that overlap graphs have “good” cut-covers, answering an open question of Kaklamanis, Krizanc and Rao (1993). Consequently, overlap graphs can be emulated on hypercube graphs with a constant factor of slow-down and on butterfly graphs with a factor of O(log^* n) slow-down. Therefore, computations on overlap graphs, such as finite element and finite difference methods on “well-conditioned” meshes and image processing on k-nearest neighbor graphs, can be performed on hypercubic parallel machines with a linear speed-up. Our result, in conjunction with a result of Plotkin, Rao and Smith, also yields a combinatorial proof that overlap graphs have separators of sublinear size. We also show that with high probability, the Delaunay diagram, the relative neighborhood graph, and the k-nearest neighbor graph of a random point set exclude K_h as a depth L minor if h = Ω(L^d/2 log n).     

Combinatorial curvature for planar graphs

Regarding an infinite planar graph G as a discrete analogue of a noncompact simply connected Riemannian surface, we introduce the combinatorial curvature of G corresponding to the sectional curvature of a manifold. We show this curvature has the property that its negative values are bounded above by a universal negative constant. We also prove that G is hyperbolic if its curvature is negative. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 220–229, 2001     

系列平行图上带时间约束的Steiner最小树问题

对一类特殊系列平行图上带有时间约束的Steiner最小树问题,证明了其复杂性为NPC,并给出了一个完全多项式时间近似方案.     

Combinatorial modulus and type of graphs

Let A be the 1-skeleton of a triangulated topological annulus. We establish bounds on the combinatorial modulus of a refinement A^′, formed by attaching new vertices and edges to A, that depend only on the refinement and not on the structure of A itself. This immediately applies to showing that a disk triangulation graph may be refined without changing its combinatorial type, provided the refinement is not too wild. We also explore the type problem in terms of disk growth, proving a parabolicity condition based on a superlinear growth rate, which we also prove optimal. We prove our results with no degree restrictions in both the EEL and VEL settings and examine type problems for more general complexes and dual graphs.