Smallest close to regular bipartite graphs without an almost perfect matching

A graph G is close to regular or more precisely a （d, d ＋ k）-graph, if the degree of each vertex of G is between d and d ＋ k. Let d ≥ 2 be an integer, and let G be a connected bipartite （d, d＋k）-graph with partite sets X and Y such that ｜X｜- ｜Y｜＋1. If G is of order n without an almost perfect matching, then we show in this paper that·n ≥ 6d ＋7 when k = 1,·n ≥ 4d＋ 5 when k = 2,·n ≥ 4d＋3 when k≥3.Examples will demonstrate that the given bounds on the order of G are the best possible.     

Longest cycles in k-connected graphs with given independence number

The Chvátal–Erd?s Theorem states that every graph whose connectivity is at least its independence number has a spanning cycle. In 1976, Fouquet and Jolivet conjectured an extension: If G is an n-vertex k-connected graph with independence number a, and a?k, then G has a cycle with length at least . We prove this conjecture.     

Toughness and the existence ofk-factors. II

In a paper with the same title [3], we proved Chvátal's conjecture thatk-tough graphs havek-factors if they satisfy trivial necessary conditions. In this paper, we prove the following stronger result: Suppose|V(G)| k + 1,k |V(G)| even, and|S| k w(G – S) – 7/8k ifw(G – S) 2, wherew(G – S) is the number of connected components ofG – S. ThenG has ak-factor.     

Paths with restricted degrees of their vertices in planar graphs

In this paper it is proved that every 3-connected planar graph contains a path on 3 vertices each of which is of degree at most 15 and a path on 4 vertices each of which has degree at most 23. Analogous results are stated for 3-connected planar graphs of minimum degree 4 and 5. Moreover, for every pair of integers n 3, k 4 there is a 2-connected planar graph such that every path on n vertices in it has a vertex of degree k.     

Graphs S(n, k) and a Variant of the Tower of Hanoi Problem

For any n 1 and any k 1, a graph S(n, k) is introduced. Vertices of S(n, k) are n-tuples over {1, 2,. . . k} and two n-tuples are adjacent if they are in a certain relation. These graphs are graphs of a particular variant of the Tower of Hanoi problem. Namely, the graphs S(n, 3) are isomorphic to the graphs of the Tower of Hanoi problem. It is proved that there are at most two shortest paths between any two vertices of S(n, k). Together with a formula for the distance, this result is used to compute the distance between two vertices in O(n) time. It is also shown that for k 3, the graphs S(n, k) are Hamiltonian.     

Bounds on the Chvátal Rank of Polytopes in the 0/1-Cube*

Gomorys and Chvátals cutting-plane procedure proves recursively the validity of linear inequalities for the integer hull of a given polyhedron. The Chvátal rank of the polyhedron is the number of rounds needed to obtain all valid inequalities. It is well known that the Chvátal rank can be arbitrarily large, even if the polyhedron is bounded, if it is 2-dimensional, and if its integer hull is a 0/1-polytope.We show that the Chvátal rank of polyhedra featured in common relaxations of many combinatorial optimization problems is rather small; in fact, we prove that the rank of every polytope contained in the n-dimensional 0/1-cube is at most n ² (1+log n). Moreover, we also demonstrate that the rank of any polytope in the 0/1-cube whose integer hull is defined by inequalities with constant coefficients is O(n).Finally, we provide a family of polytopes contained in the 0/1-cube whose Chvátal rank is at least (1 + ) n, for some > 0.* An extended abstract of this paper appeared in the Proceedings of the 7th International Conference on Integer Programming and Combinatorial Optimization [20].     

Maximal Total Length Of k Disjoint Cycles In Bipartite Graphs

Let k, s and n be three integers with sk2, n2k+1. Let G=(V ₁,V ₂;E) be a bipartite graph with |V ₁|=|V ₂|=n. If the minimum degree of G is at least s+1, then G contains k vertex-disjoint cycles covering at least min(2n,4s) vertices of G.     

Hadwiger Number and the Cartesian Product of Graphs

The Hadwiger number η(G) of a graph G is the largest integer n for which the complete graph K _n on n vertices is a minor of G. Hadwiger conjectured that for every graph G, η(G) ≥ χ(G), where χ(G) is the chromatic number of G. In this paper, we study the Hadwiger number of the Cartesian product of graphs. As the main result of this paper, we prove that for any two graphs G ₁ and G ₂ with η(G ₁) = h and η(G ₂) = l. We show that the above lower bound is asymptotically best possible when h ≥ l. This asymptotically settles a question of Z. Miller (1978). As consequences of our main result, we show the following:

Edge guards for the fortress problem

The fortress problem is one of determining a set of guards to cover the exterior of a simple polygon. O'Rourke and Wood [cited in 7] showed that vertex guards are sometimes necessary and always sufficient for an -vertex polygon. In this paper, we solve the same problem for edge guards. Tight bounds of and edge guards are obtained for general and rectilinear polygons, respectively. Received 19 June 1999; in revised form 23 March 2000.     

An Analogue of Hajós' Theorem for the Circular Chromatic Number (II)

This paper designs a set of graph operations, and proves that for 2k/d<3, starting from K_k/_d, by repeatedly applying these operations, one can construct all graphs G with _c(G)k/d. Together with the result proved in [20], where a set of graph operations were designed to construct graphs G with _c(G)k/d for k/d3, we have a complete analogue of Hajós' Theorem for the circular chromatic number. This research was partially supported by the National Science Council under grant NSC 89-2115-M-110-003     

On distance-regular graphs withk i =k j,II

Let be a distance-regular graph of diameterd andi-th valencyk _i. We show that ifk ₂ = k_j for 2 +j d and 2 <j, then is a polygon (k = 2) or an antipodal 2-cover (k _d = 1). We also give a short proof of Terwilliger's inequality for bipartite distance-regular graphs and a refinement of Ivanov's argument on diameter bound.     

Graphs isomorphic to subgraphs of their iterated line graphs

A graphG is said to be embeddable into a graphH, if there is an isomorphism ofG into a subgraph ofH. It is shown in this paper that every unicycle or tree which is neither a path norK _1,3 embeds in itsn-th iterated line graph forn1 or 2, 3, and that every other connected graph that embeds in itsn-th iterated line graph may be constructed from such an embedded unicycle or tree in a natural way. A special kind of embedding of graph into itsn-th iterated line graph, called incidence embedding, is studied. Moreover, it is shown that for every positive integerk, there exists a graphG such that (G) = , where (G) is the leastn1 for whichG embeds inL ⁿ(G).     

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.     

Longest Cycles in 3-connected Graphs with Given Independence Number

Fouquet and Jolivet conjectured that a k-connected graph of order n and independence number α ≥ k has a cycle of length at least [Fouquet and Jolivet, Problèmes combinatoires et théorie des graphes Orsay (1976), Problems, page 438]. Here we prove this conjecture for k=3.     

Multiple-guard kernels of simple polygons

We study a generalization of the kernel of a polygon. A polygonP isk guardable if there arek points inP such that, for all pointsp inP, there is at least one of thek points that seesp. We call thek points ak-guard set ofP and thek-kernel of a polygonP is the union of allk-guard sets ofP. The usual definition of the kernel of a polygon is now the one-kernel in this notation.We show that the two-kernel of a simple polygonP withn edges has O(n⁴) components and that there are polygons whose two-kernel has (n) components. Moreover, we show that the components of the two-kernel of a simple polygon can be paired in a natural manner which implies that the two-kernel of a simple polygon has either one component or an even number of components. Finally, we consider the question of whether there is a non-star-shaped simple polygonP such that 2-kernel(P) = P. We show that when the two-kernel has only one component, it contains a hole; hence, the two-kernel of a simple polygonP is neverP itself. For everyk 1, there are, however, polygonsP _k with holes such thatk-kernel(P_k) = P_k.This work was supported under grant No. Ot 64/8-1, Diskrete Probleme from the the Deutsche Forschungsgemeinschaft, grants from the Natural Sciences and Engineering Council of Canada, from the Information Technology Research Centre of Ontario, and from the Research Grants Council of Hong Kong.     

On the number of nowhere zero points in linear mappings

LetA be a nonsingularn byn matrix over the finite fieldGF _q ,k=n/2,q=p ^a ,a1, wherep is prime. LetP(A,q) denote the number of vectorsx in (GF _q ) ⁿ such that bothx andAx have no zero component. We prove that forn2, and ,P(A,q)[(q–1)(q–3)] ^k (q–2) ^n–2k and describe all matricesA for which the equality holds. We also prove that the result conjectured in [1], namely thatP(A,q)1, is true for allqn+23 orqn+14.     

Translativity of absolute weighted mean summability

In this paper, we give necessary and sufficient conditions on (p _n) for |R,p _n| _k , k 1, to be translative. So we extend the known results of Al-Madi [1] and Cesco [4] to the case k > 1.     

On the order of certain close to regular graphs without a matching of given size

A graph G is a {d, d+k}-graph, if one vertex has degree d+k and the remaining vertices of G have degree d. In the special case of k = 0, the graph G is d-regular. Let k, p ⩾ 0 and d, n ⩾ 1 be integers such that n and p are of the same parity. If G is a connected {d, d+k{-graph of order n without a matching M of size 2|M| = n − p, then we show in this paper the following: If d = 2, then k ⩾ 2(p + 2) and

Contractible edges in non-separating cycles

An edge of ak-connected graph is said to bek-contractible if the contraction of the edge results in ak-connected graph. We prove that every triangle-freek-connected graphG has an induced cycleC such that all edges ofC arek-contractible and such thatG–V(C) is (k–3)-connected (k4). This result unifies two theorems by Thomassen [5] and Egawa et. al. [3].Dedicated to Professor Toshiro Tsuzuku on his sixtieth birthday