Myriam Preissmann

On the NP-completeness of the k -colorability problem for triangle-free graphs

Abstract We show that the question “Is a graph 3-colorable?” remains NP-complete when restricted to the class of triangle-free graphs with maximum degree 4. Likewise the question “Is a triangle-free graph k -colorable?” is shown to be NP-complete for any fixed value of k ⩾ 4.

Linear recognition of pseudo-split graphs

We characterize the graphs with no induced 2K2 or C4 and present a linear-time recognition algorithm.

On Minimally Imperfect Graphs with Circular Symmetry

Let G be a connected graph on n vertices. A spanning tree T of G is called an independence tree, if the set of end vertices of T (vertices with degree one in T) is an independent set in G. If G has an independence tree, then α t(G) denotes the maximum number of end vertices of an independence tree of G. We show that determining αt of a graph is an NP-hard problem. We give the following analogue of a well-known result due to Chvatal and Erdos. If αt(G) ≤ κ(G) - 1, then G is hamiltonian. We show that the condition is sharp. An I≤k-tree of G is an independence tree of G with at most k end vertices or a Hamilton cycle of G. We prove the following two generalizations of a theorem of Ore. If G has an independence tree T with k end vertices such that two end vertices of T have degree sum at least n - k + 2 in G, then G has an I≤k-1-tree. If the degree sum of each pair of nonadjacent vertices of G is at least n - k + 1, then G has an I≤k-tree. Finally, we prove the following analogue of a closure theorem due to Bondy and Chvatal. If the degree sum of two nonadjacent vertices u and v of G is at least n - 1, then G has an I≤k-tree if and only if G + uv has an I≤k-tree (k ≥ 2). The last three results are all best possible with respect to the degree sum conditions.

Locally perfect graphs

Abstract A new property of minimal imperfect graphs is given. This leads to a way to add a new vertex to a perfect graph so that the resulting graph remains perfect. It is shown that the same holds for strongly perfect graphs. Some colouring properties are also considered. In particular we define a new kind of colouring, the locally perfect colouring, and we show that the graphs having such a colouring for any subgraph are perfect. Such graphs are called locally perfect. The triangulated graphs, the parity graphs, and the perfect graphs without a clique with more than three vertices are shown to be locally perfect. Finally the graphs obtained by some rules of construction are studied.

Imperfect and Nonideal Clutters: A Common Approach

We prove three theorems. First, Lovász’s theorem about minimal imperfect clutters, including also Padberg’s corollaries. Second, Lehman’s result on minimal nonideal clutters. Third, a common generalization of these two. The endeavor of working out a ‘common denominator’ for Lovász’s and Lehman’s theorems leads, besides the common generalization, to a better understanding and simple polyhedral proofs of both.

A charming class of perfectly orderable graphs

Abstract We investigate the following conjecture of Vasek Chvatal: any weakly triangulated graph containing no induced path on five vertices is perfectly orderable. In the process we define a new polynomially recognizable class of perfectly orderable graphs called charming. We show that every weakly triangulated graph not containing as an induced subgraph a path on five vertices or the complement of a path on six vertices is charming.

Generic algorithms for some decision problems on fasciagraphs and rotagraphs

Abstract A fasciagraph consists of a sequence of copies of the same graph, each copy being linked to the next one according to a regular scheme. More precisely, a fasciagraph is characterized by an integer n (the number of copies or fibers) and a mixed graph M . In a rotagraph, the last copy is also linked to the first one. In the literature, similar methods were used to address various problems on rotagraphs and fasciagraphs. The goal of our work is to define a class of decision problems for which this kind of method works. For this purpose, we introduce the notion of pseudo- d -local q -properties of fasciagraphs and rotagraphs. For a mixed graph M and a pseudo- d -local q -property P , we propose a generic algorithm for rotagraphs (respectively, fasciagraphs) that computes in one run the data that allow one to decide, for any integer n ≥ d (respectively, n ≥ d + 2 ), whether the rotagraph (respectively, fasciagraph) of length n based on M satisfies P , using only a small number of elementary operations independent of n .

Sequential colorings and perfect graphs

We study a variant of a sequential algorithm for coloring the vertices of a graph, using bichromatic exchanges, and exhibit a class of graphs which have the property that there is an ordering of the vertices such that this algorithm provides an optimal coloring for each induced ordered subgraph. These graphs are perfect and generalize several well-known classes of perfect graphs such as line-graphs of bipartite graphs or triangulated graphs.

Split-Neighborhood Graphs and the Strong Perfect Graph Conjecture

We introduce the class of graphs such that every induced subgraph possesses a vertex whose neighbourhood can be split into a clique and a stable set. We prove that this class satisfies Berge?s strong perfect graph conjecture. This class contains several well-known classes of (perfect) graphs and is polynomially recognizable.

Graphs with largest number of minimum cuts

Abstract Let σ(n, k) be the largest number of k-cuts in a k-edge-connected multigraph with n vertices. We determine σ(n, k) and characterize extremal multigraphs for every n and k. The same problem is also investigated for graphs with no multiple edges.