Rank of maximum matchings in a graph

The matching polytope is the convex hull of the incidence vectors of all (not necessarily perfect) matchings of a graphG. We consider here the problem of computing the dimension of the face of this polytope which contains the maximum cardinality matchings ofG and give a good characterization of this quantity, in terms of the cyclomatic number of the graph and families of odd subsets of the nodes which are always nearly perfectly matched by every maximum matching.This is equivalent to finding a maximum number of linearly independent representative vectors of maximum matchings ofG; the size of such a set is called thematching rank ofG. We also give in the last section a way of computing that rank independently of those parameters.Note that this gives us a good lower bound on the number of those matchings.     

The Ulam number of infinite graphs

The following is a conjecture of Ulam: In any partition of the integer lattice on the plane into uniformly bounded sets, there exists a set that is adjacent to at least six other sets. Two sets are adjacent if each contain a vertex of the same unit square. This problem is generalized as follows. Given any uniformly bounded partitionP of the vertex set of an infinite graphG with finite maximum degree, letP (G) denote the graph obtained by letting each set of the partition be a vertex ofP (G) where two vertices ofP (G) are adjacent if and only if the corresponding sets have an edge between them. The Ulam number ofG is defined as the minimum of the maximum degree ofP (G) where the minimum is taken over all uniformly bounded partitionsP. We have characterized the graphs with Ulam number 0, 1, and 2. Restricting the partitions of the vertex set to connected subsets, we obtain the connected Ulam number ofG. We have evaluated the connected Ulam numbers for several infinite graphs. For instance we have shown that the connected Ulam number is 4 ifG is an infinite grid graph. We have settled the Ulam conjecture for the connected case by proving that the connected Ulam number is 6 for an infinite triangular grid graph. The general Ulam conjecture is equivalent to proving that the Ulam number of the infinite triangular grid graph equals 6. We also describe some interesting geometric consequences of the Ulam number, mainly concerning good drawings of infinite graphs.     

Pebbling numbers of some graphs

Chung defined a pebbling move on a graphG as the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The pebbling number of a connected graphG, f(G), is the leastn such that any distribution ofn pebbles onG allows one pebble to be moved to any specified but arbitrary vertex by a sequence of pebbling moves. Graham conjectured that for any connected graphsG andH, f(G xH) ≤ f(G)f(H). In the present paper the pebbling numbers of the product of two fan graphs and the product of two wheel graphs are computed. As a corollary, Graham’s conjecture holds whenG andH are fan graphs or wheel graphs.     

Perfect graphs and norms

For a given undirected graphG=(V, ) withn vertices we define four norms on ⁿ , namely, where (resp.) stands for the family of all maximal cliques inG (resp. , the complement ofG). The goal of this note is to demonstrate the usefulness of some notions and techniques from functional analysis in graph theory by showing how Theorem 2.1 (G is -perfect if and only if the norms are equal) together with the reflexivity of the space ⁿ equipped with either of the norms above easily yield one new result (Theorem 2.2) and two known characterizations of perfect graphs (Theorems 2.3–2.4). Namely, Theorem 2.2 provides a characterization of -perfection that is strongly related to that of Lovász (1972). It implies that the Lovász inequality is exactly the classical Schwartz inequality for the space ( ⁿ , ·) restricted to (0, 1) vectorsx, y satisfyingx = y. Theorem 2.3 is well known as the Perfect Graph Theorem, while Theorem 2.4, due to V. Chvátal and D.R. Fulkerson, characterizes -perfection of a graphG in terms of the equality between the vertex packing polytope ofG and the fractional vertex packing polytope ofG.     

Facets of the balanced (acyclic) induced subgraph polytope

A signed graph is a graph whose edges are labelled positive or negative. A signed graph is said to be balanced if the set of negative edges form a cut. The balanced induced subgraph polytopeP(G) of a graphG is the convex hull of the incidence vectors of all node sets that induce balanced subgraphs ofG. In this paper we exhibit various (rank) facet defining inequalities. We describe several methods with which new facet defining inequalities ofP(G) can be constructed from known ones. Finding a maximum weighted balanced induced subgraph of a series parallel graph is a polynomial problem. We show that for this class of graphsP(G) may have complicated facet defining inequalities. We derive analogous results for the polytope of acyclic induced subgraphs.Research supported in part by the Natural Sciences and Engineering Research Council of Canada; the second author has also been supported by C.P. Rail.     

On toughness and (g, f)-factors in bipartite graphs

Let G be a bipartite graph andg and f be two positive integer-valued functions defined on vertex setV(G) ofG such thatg(x) ≤ f(x) for anyx ? V(G). In this paper, a new isolated toughness ofG is defined and some sufficient conditions related to the new toughness forG to have (g,f )-factors are obtained. Furthermore, these results are proved to be sharp in some sense.     

On σ-equivalence and χ-equivalence of graphs

We define a partial ordering on the set of σ-polynomials as well as a vertex splitting operation on the set of graphs, and introduce the notions of σ-equivalence and σ-uniqueness of graphs. Let σ(G) be the σ-polynomial of a graph G and (OVERBAR)σ(G) = σ(G^c). Let H = (G, v, A, B) be a vertex splitting graph of G. We prove that (OVERBAR)σ(G) ≤ (OVERBAR)σ(H) and the equality holds if and only if every vertex of A is adjacent to every vertex of B. This gives us an effective means to find σ-equivalent and χ-equivalent graphs. A necessary and sufficient condition for a graph to be χ-unique but not σ-unique is also obtained. © 1996 John Wiley & Sons, Inc.     

Cycle reversals in oriented plane quadrangulations and orthogonal plane partitions

Aplane quadrangulation G is a simple plane graph such that each face ofG is quadrilateral. A (^*) -orientation D ^*(G) ofG is an orientation ofG such that the outdegree of each vertex on G is 1 and the outdegrees of other vertices are all 2, where G denotes the outer 4-cycle ofG. In this paper, we shall show that every plane quadrangulationG has at least one (^*)-orientation. We also show that any two (*)-orientations ofG can be transformed into one another by a sequence of 4-cycle reversals. Moreover, we apply this fact toorthogonal plane partitions, which are partitions of a square into rectangles by straight segments.A research fellow of the Japan Society for the Promotion of Science.     

On semisymmetric cubic graphs of order 6p~2

A graph r is said to be G-semisymmetric if it is regular and there exists a subgroup G of A := Aut(Г) acting transitively on its edge set but not on its vertex set. In the case of G. = A, we call r a semisymmetric graph. The aim of this paper is to investigate (G-)semisymmetric graphs of prime degree. We give a group-theoretical construction of such graphs, and give a classification of semisymmetric cubic graphs of order 6p2 for an odd prime p.     

Some topics in cochromatic theory

GivenG, a graph, the cochromatic number,Z(G), ofG is the fewest number of sets into which the vertex set can be partitioned so that each set induces a complete or an empty graph. A graph is critically cochromatic if the removal of any of its vertices decreases its cochromatic number. A graph is uniquely cochromatic if there is exactly one partition of minimum order in which each set induces a complete or an empty graph. A graph is comaximal if the removal of any edge increases its cochromatic number. These and related concepts are examined.     

Relaxations of vertex packing

A polynomially computable upper bound for the weighted independence number of a graph is studied. This gives rise to a convex body containing the vertex packing polytope of the graph. This body is a polytope if and only if the graph is perfect. As an application of these notions, we show that the maximum weight independent set of an h-perfect graph can be found in polynomial time.     

(g, f)-factorizations orthogonal to a subgraph in graphs

LetG be a graph with vertex setV (G) and edge setE (G), and letg andf be two integer-valued functions defined on V(G) such thatg(x)⩽(x) for every vertexx ofV(G). It was conjectured that ifG is an (mg +m - 1,mf -m+1)-graph andH a subgraph ofG withm edges, thenG has a (g,f)-factorization orthogonal toH. This conjecture is proved affirmatively. Project supported by the National Natural Science Foundation of China.     

Uniquely cocolourable graphs

A graphG is uniquelyn-cocolourable if (its cochromatic number)z(G) = n and alln-cocolourings ofG induce the same partition of its vertex set. We prove that there are infinitely many uniquelyn-cocolourable graphs and that the numbers of complete and empty colour classes can be prescribed. It is also shown that everyn-cocolourable graph is an induced subgraph of a uniquelyn-cocolourable graph.     

Connected Domination Number of a Graph and its Complement

A set S of vertices in a graph G is a connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraph induced by S is connected. The connected domination number γ _c (G) is the minimum size of such a set. Let d^*(G)=min{d(G),d([`(G)])}{\delta^*(G)={\rm min}\{\delta(G),\delta({\overline{G}})\}} , where [`(G)]{{\overline{G}}} is the complement of G and δ(G) is the minimum vertex degree. We prove that when G and [`(G)]{{\overline{G}}} are both connected, g<sub>c</sub>(G)+g<sub>c</sub>([`(G)]) ￡ d^*(G)+4-(g_c(G)-3)(g_c([`(G)])-3){{\gamma_c}(G)+{\gamma_c}({\overline{G}})\le \delta^*(G)+4-({\gamma_c}(G)-3)({\gamma_c}({\overline{G}})-3)} . As a corollary, g<sub>c</sub>(G)+g<sub>c</sub>([`(G)]) ￡ \frac3n4{{\gamma_c}(G)+{\gamma_c}({\overline{G}})\le \frac{3n}{4}} when δ*(G) ≥ 3 and n ≥ 14, where G has n vertices. We also prove that g_c(G)+g_c([`(G)]) ￡ d<sup>*</sup>(G)+2{{\gamma_c}(G)+{\gamma_c}({\overline{G}})\le \delta^*(G)+2} when g<sub>c</sub>(G),g<sub>c</sub>([`(G)]) 3 4{{\gamma_c}(G),{\gamma_c}({\overline{G}})\ge 4} . This bound is sharp when δ*(G) = 6, and equality can only hold when δ*(G) = 6. Finally, we prove that g_c(G)g_c([`(G)]) ￡ 2n-4{{\gamma_c}(G){\gamma_c}({\overline{G}})\le 2n-4} when n ≥ 7, with equality only for paths and cycles.     

Excluding Induced Subdivisions of the Bull and Related Graphs

For any graph H, let Forb*(H) be the class of graphs with no induced subdivision of H. It was conjectured in [J Graph Theory, 24 (1997), 297–311] that, for every graph H, there is a function f_H: ?→? such that for every graph G∈Forb*(H), χ(G)≤f_H(ω(G)). We prove this conjecture for several graphs H, namely the paw (a triangle with a pendant edge), the bull (a triangle with two vertex‐disjoint pendant edges), and what we call a “necklace,” that is, a graph obtained from a path by choosing a matching such that no edge of the matching is incident with an endpoint of the path, and for each edge of the matching, adding a vertex adjacent to the ends of this edge. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:49–68, 2012     

The perfectly Matchable Subgraph Polytope of an arbitrary graph

The Perfectly Matchable Subgraph Polytope of a graphG=(V, E), denoted byPMS(G), is the convex hull of the incidence vectors of thoseXV which induce a subgraph having a perfect matching. We describe a linear system whose solution set isPMS(G), for a general (nonbipartite) graphG. We show how it can be derived via a projection technique from Edmonds' characterization of the matching polytope ofG. We also show that this system can be deduced from the earlier bipartite case [2], by using the Edmonds-Gallai structure theorem. Finally, we characterize which inequalities are facet inducing forPMS(G), and hence essential.     

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).     

On the set covering polytope: I. All the facets with coefficients in {0, 1, 2}

While the set packing polytope, through its connection with vertex packing, has lent itself to fruitful investigations, little is known about the set covering polytope. We characterize the class of valid inequalities for the set covering polytope with coefficients equal to 0, 1 or 2, and give necessary and sufficient conditions for such an inequality to be minimal and to be facet defining. We show that all inequalities in the above class are contained in the elementary closure of the constraint set, and that 2 is the largest value ofk such that all valid inequalities for the set covering polytope with integer coefficients no greater thank are contained in the elementary closure. We point out a connection between minimal inequalities in the class investigated and certain circulant submatrices of the coefficient matrix. Finally, we discuss conditions for an inequality to cut off a fractional solution to the linear programming relaxation of the set covering problem and to improve the lower bound given by a feasible solution to the dual of the linear programming relaxation.Research supported by the National Science Foundation through grant ECS-8503198 and the Office of Naval Research through contract N0001485-K-0198.     

UPPER BOUNDS ON THE LOWER OPEN PACKING NUMBER OF A TREE

Abstract

Let G be a graph and let v be a vertex of G. The open neigbourhood N(v) of v is the set of all vertices adjacent with v in G. An open packing of G is a set of vertices whose open neighbourhoods are pairwise disjoint. The lower open packing number of G, denoted ρ^° _L(G), is the minimum cardinality of a maximal open packing of G while the (upper) open packing number of G, denoted ρ^°(G), is the maximum cardinality among all open packings of G. It is known (see [7]) that if G is a connected graph of order n ≥3, then ρ^°(G) ≤ 2n/3 and this bound is sharp (even for trees). As a consequence of this result, we know that ρ^° _L(G) ≤ 2n/3. In this paper, we improve this bound when G is a tree. We show that if G is a tree of order n with radius 3, then ρ^° _L(G) ≤ n/2 + 2 √n-1, and this bound is sharp, while if G is a tree of order n with radius at least 4, then ρ^° _L(G) is bounded above by 2n/3—O√n).     

A characterization of well-covered graphs that contain neither 4- nor 5-cycles

A graph is well covered if every maximal independent set has the same cardinality. A vertex x, in a well-covered graph G, is called extendable if G – {x} is well covered and β(G) = β(G – {x}). If G is a connected, well-covered graph containing no 4- nor 5-cycles as subgraphs and G contains an extendable vertex, then G is the disjoint union of edges and triangles together with a restricted set of edges joining extendable vertices. There are only 3 other connected, well-covered graphs of this type that do not contain an extendable vertex. Moreover, all these graphs can be recognized in polynomial time.