Star arboricity

Astar forest is a forest all of whose components are stars. Thestar arboricity, st(G) of a graphG is the minimum number of star forests whose union covers all the edges ofG. Thearboricity, A(G), of a graphG is the minimum number of forests whose union covers all the edges ofG. Clearlyst(G)A(G). In fact, Algor and Alon have given examples which show that in some casesst(G) can be as large asA(G)+(log) (where is the maximum degree of a vertex inG). We show that for any graphG, st(G)A(G)+O(log).     

Reconstructing graphs as subsumed graphs of hypergraphs,and some self-complementary triple systems

LetV be a set ofn elements. The set of allk-subsets ofV is denoted . Ak-hypergraph G consists of avertex-set V(G) and anedgeset , wherek≥2. IfG is a 3-hypergraph, then the set of edges containing a given vertexvεV(G) define a graphG _v . The graphs {G _v νvεV(G)} aresubsumed byG. Each subsumed graphG _v is a graph with vertex-setV(G) − v. They can form the set of vertex-deleted subgraphs of a graphH, that is, eachG _v ≽H −v, whereV(H)=V(G). In this case,G is a hypergraphic reconstruction ofH. We show that certain families of self-complementary graphsH can be reconstructed in this way by a hypergraphG, and thatG can be extended to a hypergraphG ^*, all of whose subsumed graphs are isomorphic toH, whereG andG ^* are self-complementary hypergraphs. In particular, the Paley graphs can be reconstructed in this way. This work was supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada.     

Optimization and optimality test for the Max-Cut Problem

We show that the following two problems are polynomially equivalent:

Solution to a problem of C. D. Godsil regarding bipartite graphs with unique perfect matching

We give the solution to the following question of C. D. Godsil[2]: Among the bipartite graphsG with a unique perfect matching and such that a bipartite graph obtains when the edges of the matching are contracted, characterize those having the property thatG ⁺G, whereG ⁺ is the bipartite multigraph whose adjacency matrix,B ⁺, is diagonally similar to the inverse of the adjacency matrix ofG put in lower-triangular form. The characterization is thatG must be obtainable from a bipartite graph by adding, to each vertex, a neighbor of degree one. Our approach relies on the association of a directed graph to each pair (G, M) of a bipartite graphG and a perfect matchingM ofG.     

Rotation numbers for complete tripartite graphs

Arooted graph is a pair (G, x), whereG is a simple undirected graph andx V(G). IfG is rooted atx, then itsrotation number h(G, x) is the minimum number of edges in a graphF of the same order asG such that for allv V(F), we can find a copy ofG inF with the rootx atv. Rotation numbers for all complete bipartite graphs are now known (see [2], [4], [7]). In this paper we calculate rotation numbers for complete tripartite graphs with rootx in the largest vertex class.Funded by the Science and Engineering Research Council.     

Isoperimetric numbers of graph bundles

The main aim of this paper is to give some upper and lower bounds for the isoperimetric numbers of graph coverings or graph bundles, with exact values in some special cases. In addition, we show that the isoperimetric number of any covering graph is not greater than that of the base graph. Mohar's theorem for the isoperimetric number of the cartesian product of a graph and a complete graph can be extended to a more general case: The isoperimetric numberi(G × K _2n) of the cartesian product of any graphG and a complete graphK _2n on even vertices is the minimum of the isoperimetric numberi(G) andn, and it is also a sharp lower bound of the isoperimetric numbers of all graph bundles over the graphG with fiberK _2n. Furthermore, ifn 2i(G) then the isoperimetric number of any graph bundle overG with fibreK _n is equal to the isoperimetric numberi(G) ofG. Partially supported by The Ministry of Education, Korea.     

The class reconstruction number of maximal planar graphs

The reconstruction numberrn(G) of a graphG was introduced by Harary and Plantholt as the smallest number of vertex-deleted subgraphsG _i =G – v _i in the deck ofG which do not all appear in the deck of any other graph. For any graph theoretic propertyP, Harary defined theP-reconstruction number of a graph G P as the smallest number of theG _i in the deck ofG, which do not all appear in the deck of any other graph inP We now study the maximal planar graph reconstruction numberrn(G), proving that its value is either 1 or 2 and characterizing those with value 1.     

Polynomial invariants of graphs II

Negami and Kawagoe has already defined a polynomial associated with each graphG as what discriminates graphs more finely than the polynomialf(G) defined by Negami and the Tutte polynomial. In this paper, we shall show that the polynomial includes potentially the generating function counting the independent sets and the degree sequence of a graphG, which cannot be recognized fromf(G) in general, and discuss on of treesT with observations by computer experiments.     

On the maximal number of certain subgraphs inK r -free graphs

Given two graphsH andG, letH(G) denote the number of subgraphs ofG isomorphic toH. We prove that ifH is a bipartite graph with a one-factor, then for every triangle-free graphG withn verticesH(G) H(T ₂(n)), whereT ₂(n) denotes the complete bipartite graph ofn vertices whose colour classes are as equal as possible. We also prove that ifK is a completet-partite graph ofm vertices,r > t, n max(m, r – 1), then there exists a complete (r – 1)-partite graphG* withn vertices such thatK(G) K(G*) holds for everyK _r -free graphG withn vertices. In particular, in the class of allK _r -free graphs withn vertices the complete balanced (r – 1)-partite graphT _r–1(n) has the largest number of subgraphs isomorphic toK _t (t < r),C ₄,K _2,3. These generalize some theorems of Turán, Erdös and Sauer.Dedicated to Paul Turán on his 80th Birthday     

Another family of chromatically unique graphs

LetP(G; ) denote the chromatic polynomial of a graphG, expressed in the variable. ThenG is said to be chromatically unique ifG is isomorphic withH for any graphH such thatP(H; ) = P(G; ). In this paper, we provide a new family of chromatically unique graphs.Research was partly supported by the University of Agriculture research grant # 50205-91-05     

Relaxed chromatic numbers of graphs

Given any family of graphsP, theP chromatic number _p (G) of a graphG is the smallest number of classes into whichV(G) can be partitioned such that each class induces a subgraph inP. We study this for hereditary familiesP of two broad types: the graphs containing no subgraph of a fixed graphH, and the graphs that are disjoint unions of subgraphs ofH. We generalize results on ordinary chromatic number and we computeP chromatic number for special choices ofP on special classes of graphs.Research supported in part by ONR Grant N00014-85K0570 and by a grant from the University of Illinois Research Board.     

Bi-complement Reducible Graphs

We introduce a new family of bipartite graphs which is the bipartite analogue of the class ofcomplement reduciblegraphs orcographs. Abi-complement reduciblegraph orbi-cographis a bipartite graphG = (W ∪ B, E) that can be reduced to single vertices by recursively bi-complementing the edge set of all connected bipartite subgraphs. Thebi-complementedgraphofGis the graph having the same vertex setW ∪ BasG, while its edge set is equal toW × B − E. The aim of this paper is to show that there exists an equivalent definition of bi-cographs by three forbidden configurations. We also propose a tree representation for this class of graphs.     

A constrained matching problem

We show that certain manpower scheduling problems can be modeled as the following constrained matching problem. Given an undirected graphG = (V,E) with edge weights and a digraphD = (V,A). AMaster/Slave-matching (MS-matching) ofG with respect toD is a matching ofG such that for each arc (u, v) A for which the nodeu is matched, the nodev is matched, too. TheMS-Matching Problem is the problem of finding a maximum-weight MS-matching. Letk(D) be the maximum size of a (weakly) connected component ofD. We prove that MS-matching is an NP-hard problem even ifG is bipartite andk(D) 3. Moreover, we show that in the relevant special case wherek(D) 2, the MS-Matching Problem can be transformed to the ordinary Matching Problem.This research was supported by Grant 03-KL7PAS-6 of the German Federal Ministry of Research and Technology.     

A Dynamic location problem for graphs

We introduce a class of optimization problems, calleddynamic location problems, involving the processing of requests that occur sequentially at the nodes of a graphG. This leads to the definition of a new parameter of graphs, called the window indexWX(G), that measures how large a window into the future is needed to solve every instance of the dynamic location problem onG optimally on-line. We completely characterize this parameter:WX(G)k if and only ifG is a weak retract of a product of complete graphs of size at mostk. As a byproduct, we obtain two (polynomially recognizable) structural characterizations of such graphs, extending a result of Bandelt.     

Bounding the number of circuits of a graph

Letc(G) denote the number of circuits of a graphG. In this paper, we characterize those minor-closed classesG of graphs for which there is a polynomial functionp(.) such thatc(G)p(|E(G)|) for all graphsG inG.     

Removable edges in cyclically 4-edge-connected cubic graphs

We consider various ways of obtaining smaller cyclically 4-edge-connected cubic graphs from a given such graph. In particular, we consider removable edges: an edgee of a cyclically 4-edge-connected cubic graphG is said to be removable ifG is also cyclically 4-edge-connected, whereG is the cubic graph obtained fromG by deletinge and suppressing the two vertices of degree 2 created by the deletion. We prove that any cyclically 4-edge-connected cubic graphG with at least 12 vertices has at least 1/5(|E(G)| + 12) removable edges, and we characterize the graphs with exactly 1/5(|E(G)| + 12) removable edges.This work was carried out while the first author held a Niels Bohr Fellowship from the Royal Danish Academy of Sciences.     

Large induced degenerate subgraphs

A graphH isd-degenerate if every subgraph of it contains a vertex of degree smaller thand. For a graphG, let _d (G) denote the maximum number of vertices of an inducedd-degenerate subgraph ofG. Sharp lowers bounds for _d (G) in terms of the degree sequence ofG are obtained, and the minimum number of edges of a graphG withn vertices and ₂ (G) m is determined precisely for allm n. Research supported in part by an Allon Fellowship and by a Bat-Sheva de Rothschild grant.Research supported in part by NSF grant MCS 8301867, and by a Sloan Research Fellowship.     

On sparse spanners of weighted graphs

Given a graphG, a subgraphG' is at-spanner ofG if, for everyu,v V, the distance fromu tov inG' is at mostt times longer than the distance inG. In this paper we give a simple algorithm for constructing sparse spanners for arbitrary weighted graphs. We then apply this algorithm to obtain specific results for planar graphs and Euclidean graphs. We discuss the optimality of our results and present several nearly matching lower bounds.The work of G. Das and D. Joseph was supported by NSF PYI Grant DCR-8402375. The work of D. Dobkin was supported by NSF Grant CCR-8700917. The work of J. Soares was supported by CNPq proc 203039/87.4 (Brazil) and NSF Grant CCR-9014562. This research was accomplished while G. Das was a student at the University of Wisconsin-Madison. A preliminary version was presented at the Second Scandinavian Workshop on Algorithm Theory, Bergen, Norway, 1990, under the title Generating Sparse Spanners for Weighted Graphs, and proceedings appear in the series Lecture Notes in Computer Science, Springer-Verlag. The preliminary version also appears as Princeton University Technical Report CS-TR-261-90, and as University of Wisconsin-Madison Computer Sciences Technical Report 882.     

The partial complement of graphs

For a graphG, the switched graphS _v (G) ofG at a vertexv is the graph obtained fromG by deleting the edges ofG incident withv and adding the edges of incident withv. Properties of graphs whereS _v (G) G or are studied. This concept is extended to the partial complementS _H (G) where H . The investigation here centers around the existence of setsH for whichS _H (G) G. A parameter is introduced which measures how near a graph is to being self-complementary.     

On a connection between the existence ofk-trees and the toughness of a graph

A graphG has toughnesst(G) ift(G) is the largest numbert such that for any subsetS of the vertices ofG, the number of vertices inS is at leastt times the number of components ofG on deletion of the vertices inS, provided that there is then more than one component. Ak-tree of a connected graph is a spanning tree with maximum degree at mostk. We show here that if , withk 3, thenG has ak-tree. The notion of ak-tree generalizes the casek = 2 of a hamiltonian path, so that this result, as we discuss, may be of some interest in connection with Chvátal's conjecture that, for somet, every graph with toughness at leastt is hamiltonian.