On a graph property generalizing planarity and flatness

We introduce a topological graph parameter σ(G), defined for any graph G. This parameter characterizes subgraphs of paths, outerplanar graphs, planar graphs, and graphs that have a flat embedding as those graphs G with σ(G)≤1,2,3, and 4, respectively. Among several other theorems, we show that if H is a minor of G, then σ(H)≤σ(G), that σ(K _n )=n−1, and that if H is the suspension of G, then σ(H)=σ(G)+1. Furthermore, we show that μ(G)≤σ(G) + 2 for each graph G. Here μ(G) is the graph parameter introduced by Colin de Verdière in [2].     

Approximation algorithms via contraction decomposition

We prove that the edges of every graph of bounded (Euler) genus can be partitioned into any prescribed number k of pieces such that contracting any piece results in a graph of bounded treewidth (where the bound depends on k). This decomposition result parallels an analogous, simpler result for edge deletions instead of contractions, obtained in [4,20, 10, 17], and it generalizes a similar result for “compression” (a variant of contraction) in planar graphs [29]. Our decomposition result is a powerful tool for obtaining PTASs for contraction-closed problems (whose optimal solution only improves under contraction), a much more general class than minor-closed problems. We prove that any contraction-closed problem satisfying just a few simple conditions has a PTAS in bounded-genus graphs. In particular, our framework yields PTASs for the weighted Traveling Salesman Problem and for minimum-weight c-edge-connected submultigraph on bounded-genus graphs, improving and generalizing previous algorithms of [24, 1, 29, 25, 8, 5]. We also highlight the only main difficulty in extending our results to general H-minor-free graphs.     

The chromatic number of the product of two 4-chromatic graphs is 4

For any graphG and numbern≧1 two functionsf, g fromV(G) into {1, 2, ...,n} are adjacent if for all edges (a, b) ofG, f(a) ≠g(b). The graph of all such functions is the colouring graph ℒ(G) ofG. We establish first that χ(G)=n+1 implies χ(ℒ(G))=n iff χ(G ×H)=n+1 for all graphsH with χ(H)≧n+1. Then we will prove that indeed for all 4-chromatic graphsG χ(ℒ(G))=3 which establishes Hedetniemi’s [3] conjecture for 4-chromatic graphs. This research was supported by NSERC grant A7213     

Cycles in weighted graphs

A weighted graph is one in which each edgee is assigned a nonnegative numberw(e), called the weight ofe. The weightw(G) of a weighted graphG is the sum of the weights of its edges. In this paper, we prove, as conjectured in [2], that every 2-edge-connected weighted graph onn vertices contains a cycle of weight at least 2w(G)/(n–1). Furthermore, we completely characterize the 2-edge-connected weighted graphs onn vertices that contain no cycle of weight more than 2w(G)/(n–1). This generalizes, to weighted graphs, a classical result of Erds and Gallai [4].     

Planar Ramsey numbers for cycles

For two given graphs G and H the planar Ramsey number PR(G,H) is the smallest integer n such that every planar graph F on n vertices either contains a copy of G or its complement contains a copy H. By studying the existence of subhamiltonian cycles in complements of sparse graphs, we determine all planar Ramsey numbers for pairs of cycles.     

Tutte sets in graphs II: The complexity of finding maximum Tutte sets

A well-known formula of Tutte and Berge expresses the size of a maximum matching in a graph G in terms of what is usually called the deficiency. A subset X of V(G) for which this deficiency is attained is called a Tutte set of G. While much is known about maximum matchings, less is known about the structure of Tutte sets. We explored the structural aspects of Tutte sets in another paper. Here, we consider the algorithmic complexity of finding Tutte sets in a graph. We first give two polynomial algorithms for finding a maximal Tutte set. We then consider the complexity of finding a maximum Tutte set, and show it is NP-hard for general graphs, as well as for several interesting restricted classes such as planar graphs. By contrast, we show we can find maximum Tutte sets in polynomial time for graphs of level 0 or 1, elementary graphs, and 1-tough graphs.     

赋权图的w密度和w平衡性

The notion of w-density for the graphs with positive weights on vertices and nonnegative weights on edges is introduced. A weighted graph is called w-balanced if its w-density is no less than the w-density of any subgraph of it. In this paper,a good characterization of w-balanced weighted graphs is given. Applying this characterization ,many large w-balanced weighted graphs are formed by combining smaller ones. In the case where a graph is not w-balanced,a polynomial-time algorithm to find a subgraph of maximum w-density is proposed. It is shown that the w-density theory is closely related to the study of SEW(G,w) games.     

Almost 2-Homogeneous Graphs and Completely Regular Quadrangles

Many known distance-regular graphs have extra combinatorial regularities: One of them is t-homogeneity. A bipartite or almost bipartite distance-regular graph is 2-homogeneous if the number γ _i  = |{x | ∂(u, x) = ∂(v, x) = 1 and ∂(w, x) = i − 1}| (i = 2, 3,..., d) depends only on i whenever ∂(u, v) = 2 and ∂(u, w) = ∂(v, w) = i. K. Nomura gave a complete classification of bipartite and almost bipartite 2-homogeneous distance-regular graphs. In this paper, we generalize Nomura’s results by classifying 2-homogeneous triangle-free distance-regular graphs. As an application, we show that if Γ is a distance-regular graph of diameter at least four such that all quadrangles are completely regular then Γ is isomorphic to a binary Hamming graph, the folded graph of a binary Hamming graph or the coset graph of the extended binary Golay code of valency 24. We also consider the case Γ is a parallelogram-free distance-regular graph. This research was partially supported by the Grant-in-Aid for Scientific Research (No.17540039), Japan Society of the Promotion of Science.     

CLASSIFICATION OF COMPLETE 5-PARTITE GRAPHS AND CHROMATICITY OF 5-PARTITE GRAPHSWITH 5n VERTICES

For a graph G,P(G,λ)denotes the chromatic polynomial of G. Two graphs G and H are said to be chromatically equivalent,denoted by G-H,if P(G,λ)=p(H,λ). Let[G]= {H|H-G}. If [G]={G},then G is said to be chromatically unique. For a complete 5-partite graph G with 5n vertices, define θ(G)=(a(G,6)-2^n 1-2^n-1 5)/2n-2,where a(G,6) denotes the number of 6-independent partitions of G. In this paper, the authors show that θ(G)≥0 and determine all graphs with θ(G)= 0, 1, 2, 5/2, 7/2, 4, 17/4. By using these results the chromaticity of 5-partite graphs of the form G-S with θ(G)=0,1,2,5/2,7/2,4,17/4 is investigated,where S is a set of edges of G. Many new chromatically unique 5-partite graphs are obtained.     

The bondage numbers of graphs with small crossing numbers

The bondage number b(G) of a nonempty graph G is the cardinality of a smallest edge set whose removal from G results in a graph with domination number greater than the domination number γ(G) of G. Kang and Yuan proved b(G)?8 for every connected planar graph G. Fischermann, Rautenbach and Volkmann obtained some further results for connected planar graphs. In this paper, we generalize their results to connected graphs with small crossing numbers.     

A bound on the treewidth of planar even-hole-free graphs

The class of planar graphs has unbounded treewidth, since the k×k grid, ∀k∈N, is planar and has treewidth k. So, it is of interest to determine subclasses of planar graphs which have bounded treewidth. In this paper, we show that if G is an even-hole-free planar graph, then it does not contain a 9×9 grid minor. As a result, we have that even-hole-free planar graphs have treewidth at most 49.     

Closed Separator Sets

A smallest separator in a finite, simple, undirected graph G is a set S ⊆ V (G) such that G–S is disconnected and |S|=κ(G), where κ(G) denotes the connectivity of G. A set S of smallest separators in G is defined to be closed if for every pair S,T ∈ S, every component C of G–S, and every component S of G–T intersecting C either X(C,D) := (V (C) ∩ T) ∪ (T ∩ S) ∪ (S ∩ V (D)) is in S or |X(C,D)| > κ(G). This leads, canonically, to a closure system on the (closed) set of all smallest separators of G. A graph H with is defined to be S-augmenting if no member of S is a smallest separator in G ∪ H:=(V (G) ∪ V (H), E(G) ∪ E(H)). It is proved that if S is closed then every minimally S-augmenting graph is a forest, which generalizes a result of Jordán. Several applications are included, among them a generalization of a Theorem of Mader on disjoint fragments in critically k-connected graphs, a Theorem of Su on highly critically k-connected graphs, and an affirmative answer to a conjecture of Su on disjoint fragments in contraction critically k-connected graphs of maximal minimum degree.     

NOTE The Johnson Graphs Satisfy a Distance Extension Property

G has property if whenever F and H are connected graphs with and |H|=|F|+1, and and are isometric embeddings, then there is an isometric embedding such that . It is easy to construct an infinite graph with for all k, and holds in almost all finite graphs. Prior to this work, it was not known whether there exist any finite graphs with . We show that the Johnson graphs J(n,3) satisfy whenever , and that J(6,3) is the smallest graph satisfying . We also construct finite graphs satisfying and local versions of the extension axioms studied in connection with the Rado universal graph. Received June 9, 1998     

Long paths in sparse random graphs

We consider random graphs withn labelled vertices in which edges are chosen independently and with probabilityc/n. We prove that almost every random graph of this kind contains a path of length ≧(1 −α(c))n where α(c) is an exponentially decreasing function ofc. Dedicated to Tibor Gallai on his seventieth birthday     

Some remarks on universal graphs

LetΓ be a class of countable graphs, and let ℱ(Γ) denote the class of all countable graphs that do not contain any subgraph isomorphic to a member ofΓ. Furthermore, letTΓ andHΓ denote the class of all subdivisions of graphs inΓ and the class of all graphs contracting to a member ofΓ, respectively. As the main result of this paper it is decided which of the classes ℱ(TK ⁿ ) and ℱ(HK ⁿ ),n≦ℵ₀, contain a universal element. In fact, for ℱ(TK ⁴)=ℱ(HK ⁴) a strongly universal graph is constructed, whereas for 5≦n≦ℵ₀ the classes ℱ(TK ⁿ ) and ℱ(HK ⁿ ) have no universal elements. Dedicated to Klaus Wagner on his 75^th birthday     

Lower bounds for the game colouring number of partial k-trees and planar graphs

This paper discusses the game colouring number of partial k-trees and planar graphs. Let col_g(PT_k) and col_g(P) denote the maximum game colouring number of partial k trees and the maximum game colouring number of planar graphs, respectively. In this paper, we prove that col_g(PT_k)=3k+2 and col_g(P)?11. We also prove that the game colouring number col_g(G) of a graph is a monotone parameter, i.e., if H is a subgraph of G, then col_g(H)?col_g(G).     

The polytope of win vectors

Imagine a graph as representing a fixture list with vertices corresponding to teams, and the number of edges joiningu andv as representing the number of games in whichu andv have to play each other. Each game ends in a win, loss, or tie and we say a vector =(w ₁,...,w _n) is awin vector if it represents the possible outcomes of the games, withw _i denoting the total number of games won by teami. We study combinatorial and geometric properties of the set of win vectors and, in particular, we consider the problem of counting them. We construct a fully polynomial randomized approximation scheme for their number in dense graphs. To do this we prove that the convex hull of the set of win vectors ofG forms an integral polymatroid and then use volume approximation techniques. Supported by the “DAAD Doktorandenstipendium des zweiten Hochschulsonderprogrammes HSPII/AUFE”. Partially supported by RAND-REC EC US030.     

Independence and coloring properties of direct products of some vertex-transitive graphs

Let α(G) and χ(G) denote the independence number and chromatic number of a graph G, respectively. Let G×H be the direct product graph of graphs G and H. We show that if G and H are circular graphs, Kneser graphs, or powers of cycles, then α(G×H)=max{α(G)|V(H)|,α(H)|V(G)|} and χ(G×H)=min{χ(G),χ(H)}.     

Contracting planar graphs to contractions of triangulations

For every graph H, there exists a polynomial-time algorithm deciding if a planar input graph G can be contracted to H. However, the degree of the polynomial depends on the size of H. We identify a class of graphs C such that for every fixed H∈C, there exists a linear-time algorithm deciding whether a given planar graph G can be contracted to H. The class C is the closure of planar triangulated graphs under taking of contractions. In fact, we prove that a graph H∈C if and only if there exists a constant cH such that if the treewidth of a graph is at least cH, it contains H as a contraction. We also provide a characterization of C in terms of minimal forbidden contractions.