The book crossing number of a graph

Let G be a graph on n vertices and m edges. The book crossing number of G is defined as the minimum number of edge crossings when the vertices of G are placed on the spine of a k-page book and edges are drawn on pages, such that each edge is contained by one page. Our main results are two polynomial time algorithms to generate near optimal drawing of G on books. The first algorithm give an O(log² n) times optimal solution, on small number of pages, under some restrictions. This algorithm also gives rise to the first polynomial time algorithm for approximating the rectilinear crossing number so that the coordinates of vertices in the plane are small integers, thus resolving a recent open question concerning the rectilinear crossing number. Moreover, using this algorithm we improve the best known upper bounds on the rectilinear crossing number. The second algorithm generates a drawing of G with O(m²/k²) crossings on k pages. This is within a constant multiplicative factor from our general lower bound of Ω(m³/n²k²), provided that m = Ψ(n²). © 1996 John Wiley & Sons, Inc.     

Embedding Planar Graphs at Fixed Vertex Locations

 Let G be a planar graph of n vertices, v ₁,…,v _n , and let {p ₁,…,p _n } be a set of n points in the plane. We present an algorithm for constructing in O(n ²) time a planar embedding of G, where vertex v _i is represented by point p _i and each edge is represented by a polygonal curve with O(n) bends (internal vertices). This bound is asymptotically optimal in the worst case. In fact, if G is a planar graph containing at least m pairwise independent edges and the vertices of G are randomly assigned to points in convex position, then, almost surely, every planar embedding of G mapping vertices to their assigned points and edges to polygonal curves has at least m/20 edges represented by curves with at least m/40³ bends. Received: May 24, 1999 Final version received: April 10, 2000     

The size Ramsey number of trees with bounded degree

The size Ramsey number r?(G, H) of graphs G and H is the smallest integer r? such that there is a graph F with r? edges and if the edge set of F is red-blue colored, there exists either a red copy of G or a blue copy of H in F. This article shows that r?(T_nd, T_nd) ? c · d² · n and c · n³ ? r?(K_n, T_nd) ? c(d)·n³ log n for every tree T_nd on n vertices. and maximal degree at most d and a complete graph K_n on n vertices. A generalization will be given. Probabilistic method is used throught this paper. © 1993 John Wiley Sons, Inc.     

平面图书式嵌入综述

图书式嵌入问题主要起源于大型集成电路(VLSI)设计和多层线路板印刷(PCBs)设计等诸多领域,有广泛的应用价值.图的书式嵌入是将图的点集排在一条直线上(书脊)且将边嵌入到以书脊为边界的半平面上(页)使得同页中的边互不相交.其研究的一个重要参数是页数(满足条件所需的最小页数),该问题是NP-困难的.本文主要综述平面图书式嵌入问题的相关研究.     

Contractions to k8

It is proved that the maximal number of edges in a graph with n ≧ 8 vertices that is not contractible to K₈ is 6n ? 21, unless 5 divides n, and the only graphs with n = 5m vertices and more than 6n ? 21 edges that are not contractible to K₈ are the K₅(2)-cockades that have exactly 6n ? 20 edges.     

Counting strongly‐connected,moderately sparse directed graphs

A sharp asymptotic formula for the number of strongly connected digraphs on n labelled vertices with m arcs, under the condition m ‐ n ? n^2/3, m = O(n), is obtained; this provides a partial solution of a problem posed by Wright back in 1977. This formula is a counterpart of a classic asymptotic formula, due to Bender, Canfield and McKay, for the total number of connected undirected graphs on n vertices with m edges. A key ingredient of their proof was a recurrence equation for the connected graphs count due to Wright. No analogue of Wright's recurrence seems to exist for digraphs. In a previous paper with Nick Wormald we rederived the BCM formula by counting first connected graphs among the graphs of minimum degree 2, at least. In this paper, using a similar embedding for directed graphs, we find an asymptotic formula, which includes an explicit error term, for the fraction of strongly‐connected digraphs with parameters m and n among all such digraphs with positive in/out‐degrees. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Dirac-type conditions for hamiltonian paths and cycles in 3-uniform hypergraphs

A hamiltonian path (cycle) in an n-vertex 3-uniform hypergraph is a (cyclic) ordering of the vertices in which every three consecutive vertices form an edge. For large n, we prove an analog of the celebrated Dirac theorem for graphs: there exists n₀ such that every n-vertex 3-uniform hypergraph H, n?n₀, in which each pair of vertices belongs to at least n/2−1 (⌊n/2⌋) edges, contains a hamiltonian path (cycle, respectively). Both results are easily seen to be optimal.     

Maximum hitting time for random walks on graphs

For x and y vertices of a connected graph G, let T_G(x, y) denote the expected time before a random walk starting from x reaches y. We determine, for each n > 0, the n-vertex graph G and vertices x and y for which T_G(x, y) is maximized. the extremal graph consists of a clique on ?(2n + 1)/3?) (or ?)(2_n ? 2)/3?) vertices, including x, to which a path on the remaining vertices, ending in y, has been attached; the expected time T_G(x, y) to reach y from x in this graph is approximately 4_n³/27.     

Complete subgraphs with large degree sums

Let t(n, k) denote the Turán number—the maximum number of edges in a graph on n vertices that does not contain a complete graph K_k+1. It is shown that if G is a graph on n vertices with n ≥ k²(k – 1)/4 and m < t(n, k) edges, then G contains a complete subgraph K_k such that the sum of the degrees of the vertices is at least 2km/n. This result is sharp in an asymptotic sense in that the sum of the degrees of the vertices of K_k is not in general larger, and if the number of edges in G is at most t(n, k) – ? (for an appropriate ?), then the conclusion is not in general true. © 1992 John Wiley & Sons, Inc.     

Deficient generalized Fibonacci maximum path graphs

The structure of an acyclic directed graph with n vertices and m edges, maximizing the number of distinct paths between two given vertices, is studied. In previous work it was shown that there exists such a graph containing a Hamiltonian path joining the two given vertices, thus uniquely ordering the vertices. It was further shown that such a graph contains k ? 1 full levels (an edge (i, j) belongs to level t = j ?i) and some edges of level k—a deficient k-generalized Fibonacci graph. We investigate the distribution of the edges in level 3 in a deficient 3-generalized Fibonacci graph, and develop tools that might be useful in extending the results to higher levels.     

Enumeration of Bipartite Self-Complementary Graphs

Let e(m, n), o(m, n), bsc(m, n) be the number of unlabelled bipartite graphs with an even number of edges whose partite sets have m vertices and n vertices, the number of those with an odd number of edges, and the number of unlabelled bipartite self-complementary graphs whose partite sets have m vertices and n vertices, respectively. Then this paper shows that the equality bsc(m, n) = e(m, n) ? o(m, n) holds.     

Graphs with unavoidable subgraphs with large degrees

Let ??(n, m) denote the class of simple graphs on n vertices and m edges and let G ∈ ?? (n, m). There are many results in graph theory giving conditions under which G contains certain types of subgraphs, such as cycles of given lengths, complete graphs, etc. For example, Turan's theorem gives a sufficient condition for G to contain a K_{k + 1} in terms of the number of edges in G. In this paper we prove that, for m = αn², α > (k - 1)/2k, G contains a K_{k + 1}, each vertex of which has degree at least f(α)n and determine the best possible f(α). For m = ?n²/4? + 1 we establish that G contains cycles whose vertices have certain minimum degrees. Further, for m = αn², α > 0 we establish that G contains a subgraph H with δ(H) ≥ f(α, n) and determine the best possible value of f(α, n).     

On the power sequence of a graph

Necessary and sufficient conditions for a sequence (p ₁,p ₂, …,p _n ) of positive integers to be the power sequence of a connected graph onn vertices withm edges are given. The maximum power of a connected graph onn vertices withm edges and the class of all extremal graphs are also determined.     

Embedding Generalized Petersen Graph in Books

A book embedding of a graph $G$ consists of placing the vertices of $G$ on a spine and assigning edges of the graph to pages so that edges in the same page do not cross each other. The page number is a measure of the quality of a book embedding which is the minimum number of pages in which the graph $G$ can be embedded. In this paper, the authors discuss the embedding of the generalized Petersen graph and determine that the page number of the generalized Petersen graph is three in some situations, which is best possible.     

Total domination of graphs and small transversals of hypergraphs

The main result of this paper is that every 4-uniform hypergraph on n vertices and m edges has a transversal with no more than (5n + 4m)/21 vertices. In the particular case n = m, the transversal has at most 3n/7 vertices, and this bound is sharp in the complement of the Fano plane. Chvátal and McDiarmid [5] proved that every 3-uniform hypergraph with n vertices and edges has a transversal of size n/2. Two direct corollaries of these results are that every graph with minimal degree at least 3 has total domination number at most n/2 and every graph with minimal degree at least 4 has total domination number at most 3n/7. These two bounds are sharp.     

Linear arboricity for graphs with multiple edges

Akiyama, Exoo, and Harary conjectured that for any simple graph G with maximum degree Δ(G), the linear arboricity la(G) satisfies ?Δ(G)/2? ≦ la(G) ≦ ?(Δ(G) + 1)/2?. Here it is proved that if G is a loopless graph with maximum degree Δ(G) ≦ k and maximum edge multiplicity μ(G) ≦ k ? 2ⁿ⁺¹ + 1, where k ≧ 2^n?2, then la(G) ≦ k ? 2ⁿ. It is also conjectured that for any loopless graph G, ?Δ(G)/2? ≦ la(G) ≦ ?(Δ(G) + μ(G))/2?.     

A Note on Directed Genera of Some Tournaments

An embedding of a digraph in an orientable surface is an embedding as the underlying graph and arcs in each region force a directed cycle. The directed genus is the minimum genus of surfaces in which the digraph can be directed embedded. Bonnington, Conder, Morton and McKenna [J. Combin. Theory Ser. B, 85(2002) 1-20] gave the problem that which tournaments on n vertices have the directed genus ?(n?3)(n?4)/12 ?, the genus of K_n. In this paper, we use the current graph method to show that there exists a tournament, which has the directed genus ?(n?3)(n?4)/12 ?, on n vertices if and only if n ≡ 3 or 7 (mod 12).     

On the Maximum Number of Cliques in a Graph

A clique is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for the following graph classes: (1) graphs with n vertices and m edges; (2) graphs with n vertices, m edges, and maximum degree Δ; (3) d-degenerate graphs with n vertices and m edges; (4) planar graphs with n vertices and m edges; and (5) graphs with n vertices and no K₅-minor or no K_3,3-minor. For example, the maximum number of cliques in a planar graph with n vertices is 8(n − 2). Research supported by a Marie Curie Fellowship of the European Community under contract 023865, and by the projects MCYT-FEDER BFM2003-00368 and Gen. Cat 2001SGR00224.     

An upper bound for the path number of a graph

The path number of a graph G, denoted p(G), is the minimum number of edge-disjoint paths covering the edges of G. Lovász has proved that if G has u odd vertices and g even vertices, then p(G) ≤ 1/2 u + g - 1 ≤ n - 1, where n is the total number of vertices of G. This paper clears up an error in Lovász's proof of the above result and uses an extension of his construction to show that p(G) ≤ 1/2 u + [3/4g] ≤ [3/4n].