The complexity of a class of infinite graphs

IfG _k is the family of countable graphs with nok vertex (or edge) disjoint circuits (1<k<) then there is a countableG _k G _k such that every member ofG _k is an (induced) subgraph of some member ofG _k , but no finiteG _k suffices.     

Preperfect graphs

We say that a vertexx of a graph is predominant if there exists another vertexy ofG such that either every maximum clique ofG containingy containsx or every maximum stable set containingx containsy. A graph is then called preperfect if every induced subgraph has a predominant vertex. We show that preperfect graphs are perfect, and that several well-known classes of perfect graphs are preperfect. We also derive a new characterization of perfect graphs.     

On a matroid defined by ear-decompositions of graphs

A. Frank described in [1] an algorithm to determine the minimum number of edges in a graph G whose contraction leaves a factor-critical graph and he asked if there was an algorithm for the weighted version of the problem. We prove that the minimal critical-making edge-sets form the bases of a matroid and hence the matroid greedy algorithm gives rise to the desired algorithm.Partially supported by OTKA F014919, OTKA T17181 and OTKA T17580.     

Counting perfect matchings in n-extendable graphs

The structural theory of matchings is used to establish lower bounds on the number of perfect matchings in n-extendable graphs. It is shown that any such graph on p vertices and q edges contains at least ⌈(n+1)!/4[q-p-(n-1)(2Δ-3)+4]⌉ different perfect matchings, where Δ is the maximum degree of a vertex in G.     

On kernels in perfect graphs

A kernel of a digraphD is a set of vertices which is both independent and absorbant. In 1983, C. Berge and P. Duchet conjectured that an undirected graphG is perfect if and only if the following condition is fulfilled: ifD is an orientation ofG (where pairs of opposite arcs are allowed) and if every clique ofD has a kernel thenD has a kernel. We prove here the conjecture for the complements of strongly perfect graphs and establish that a minimal counterexample to the conjecture is not a complete join of an independent set with another graph.     

The gallai-younger conjecture for planar graphs

Younger conjectured that for everyk there is ag(k) such that any digraphG withoutk vertex disjoint cycles contains a setX of at mostg(k) vertices such thatG–X has no directed cycles. Gallai had previously conjectured this result fork=1. We prove this conjecture for planar digraphs. Specifically, we show that ifG is a planar digraph withoutk vertex disjoint directed cycles, thenG contains a set of at mostO(klog(k)log(log(k))) vertices whose removal leaves an acyclic digraph. The work also suggests a conjecture concerning an extension of Vizing's Theorem for planar graphs.     

Conservative weightings and ear-decompositions of graphs

A subsetJ of edges of a connected undirected graphG=(V, E) is called ajoin if |CJ||C|/2 for every circuitC ofG. Answering a question of P. Solé and Th. Zaslavsky, we derive a min-max formula for the maximum cardinality of a joint ofG. Namely, =(+|V|–1)/2 where denotes the minimum number of edges whose contraction leaves a factor-critical graph.To study these parameters we introduce a new decomposition ofG, interesting for its own sake, whose building blocks are factor-critical graphs and matching-covered bipartite graphs. We prove that the length of such a decomposition is always and show how an optimal join can be constructed as the union of perfect matchings in the building blocks. The proof relies on the Gallai-Edmonds structure theorem and gives rise to a polynomial time algorithm to construct the optima in question.     

Generalization of matching extensions in graphs (II)

Proposed as a general framework, Liu and Yu [Generalization of matching extensions in graphs, Discrete Math. 231 (2001) 311-320.] introduced (n,k,d)-graphs to unify the concepts of deficiency of matchings, n-factor-criticality and k-extendability. Let G be a graph and let n,k and d be non-negative integers such that n+2k+d?|V(G)|-2 and |V(G)|-n-d is even. If when deleting any n vertices from G, the remaining subgraph H of G contains a k-matching and each such k-matching can be extended to a defect-d matching in H, then G is called an (n,k,d)-graph. Liu and Yu's Papee's paper, the recursive relations for distinct parameters n,k and d were presented and the impact of adding or deleting an edge also was discussed for the case d=0. In this paper, we continue the study begun by Liu and Yu and obtain new recursive results for (n,k,d)-graphs in the general case d?0.     

Pfaffian graphs, <Emphasis Type=Italic>T</Emphasis>-joins and crossing numbers

We characterize Pfaffian graphs in terms of their drawings in the plane. We generalize the techniques used in the proof of this characterization, and prove a theorem about the numbers of crossings in T-joins in different drawings of a fixed graph. As a corollary we give a new proof of a theorem of Kleitman on the parity of crossings in drawings of K _2j+1 and K _2j+1,2k+1. Partially supported by NSF grants DMS-0200595 and DMS-0701033.     

Crossing graphs of fiber-complemented graphs

Fiber-complemented graphs form a vast non-bipartite generalization of median graphs. Using a certain natural coloring of edges, induced by parallelism relation between prefibers of a fiber-complemented graph, we introduce the crossing graph of a fiber-complemented graph G as the graph whose vertices are colors, and two colors are adjacent if they cross on some induced 4-cycle in G. We show that a fiber-complemented graph is 2-connected if and only if its crossing graph is connected. We characterize those fiber-complemented graphs whose crossing graph is complete, and also those whose crossing graph is chordal.     

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.     

The classification of distance-regular graphs of type IIB

The distance-regular graphs of type IIB in Bannai and Ito [1] have intersection numbers of the form whered is the diameter of , andh, x, andt are complex constants. In this paper we show a graph of type IIB and diameterd (3d) is either the antipodal quotient of the Hamming graphH(2d+1,2), or has the same intersection numbers as the antipodal quotient ofH(2d, 2).     

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     

Cancellation properties of products of graphs

This note extends results of Fernández, Leighton, and López-Presa on the uniqueness of rth roots for disconnected graphs with respect to the Cartesian product to other products and shows that their methods also imply new cancelation laws.     

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.     

On the geodetic number of median graphs

A set of vertices S in a graph is called geodetic if every vertex of this graph lies on some shortest path between two vertices from S. In this paper, minimum geodetic sets in median graphs are studied with respect to the operation of peripheral expansion. Along the way geodetic sets of median prisms are considered and median graphs that possess a geodetic set of size two are characterized.     

Hadwiger's conjecture forK 6-free graphs

In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph ont+1 vertices ist-colourable. Whent3 this is easy, and whent=4, Wagner's theorem of 1937 shows the conjecture to be equivalent to the four-colour conjecture (the 4CC). However, whent5 it has remained open. Here we show that whent=5 it is also equivalent to the 4CC. More precisely, we show (without assuming the 4CC) that every minimal counterexample to Hadwiger's conjecture whent=5 is apex, that is, it consists of a planar graph with one additional vertex. Consequently, the 4CC implies Hadwiger's conjecture whent=5, because it implies that apex graphs are 5-colourable.Research partially supported by NSF grants number DMS 8903132, and DMS 9103480 respectively. Both authors were also partially supported by the DIMACS Center at Rutgers University, and the research was carried out partially under a consulting agreement with Bellcore.     

Routing sets in the integer lattice

A set of vertices S in a graph G is a routing set if it ensures some kind of connectivity between all pairs of vertices outside of S. Additional constraints may apply; a connected dominating set, for instance, is a special case of a routing set. We determine the size of a minimum routing set in subgraphs of the integer lattice, as well as (asymptotically) for the lattice itself.     

Independent sets ink-chromatic graphs

A graph is said to have propertyP _k if in eachk-colouring ofG using allk colours there arek independent vertices having all colours. An (unpublished) suggestion of P. Erdős is answered in the affirmative: For eachk≧3 there is a k-critical graph withP _k . With the aid of a construction of T. Gallaik-chromatic graphs (k≧7) withP _k orP _k+1 of arbitrarily high connectivity are obtained. The main result is: Eachk-chromatic graph (k≧3) of girth ≧6 hasP _k or is a circuit of length 7. Dedicated to Paul Erdős on his seventieth birthday     

On packing bipartite graphs

G andH, two simple graphs, can be packed ifG is isomorphic to a subgraph of , the complement ofH. A theorem of Catlin, Spencer and Sauer gives a sufficient condition for the existence of packing in terms of the product of the maximal degrees ofG andH. We improve this theorem for bipartite graphs. Our condition involves products of a maximum degree with an average degree. Our relaxed condition still guarantees a packing of the two bipartite graphs.the paper was written while the authors were graduate students at the University of Chicago and was completed while the first author was at M.I.T. The work of the first author was supported in part by the Air Force under Contract OSR-86-0076 and by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center-NSF-STC88-09648. The work of the second author was supported in part by NSF grant CCR-8706518.