A min-max relation for the partial q- colourings of a graph. Part II: Box perfection

AbstractThis paper examines extensions of a min-max equality (stated in C Berge, Part I) for the maximum number of nodes in a perfect graph which can be q-coloured.A system L of linear inequalities in the variables x is called TDI if for every linear function cx such that c is all integers, the dual of the linear program: maximize {cx: x satisfies L} has an integer-valued optimum solution or no optimum solution. A system L is called box TDI if L together with any inequalities l⩽x⩽u is TDI. It is a corollary of work of Fulkerson and Lov́asz that: where A is a 0–1 matrix with no all-0 column and with the 1-columns of any row not a proper subset of the 1-columns of any other row, the system L(G) = {Ax⩽1, x⩾0} is TDI if and only if A is the matrix of maximal cliques (rows) versus nodes (columns) of a perfect graph. Here we will describe a class of graphs in a graph-theoretic way, and characterize them as the graphs G for which the system L(G) is box TDI. Thus we call these graphs box perfect. We also describe some classes of box perfect graphs

Coloring vertices of a graph or finding a Meyniel obstruction

A Meyniel obstruction is an odd cycle with at least five vertices and at most one chord. A graph is Meyniel if and only if it has no Meyniel obstruction as an induced subgraph. Here we give a O(n^2) algorithm that, for any graph, finds either a clique and coloring of the same size or a Meyniel obstruction. We also give a O(n^3) algorithm that, for any graph, finds either aneasily recognizable strong stable set or a Meyniel obstruction

The travelling preacher, projection, and a lower bound for the stability number of a graph

AbstractThe coflow min–max equality is given a travelling preacher interpretation, and is applied to give a lower bound on the maximum size of a set of vertices, no two of which are joined by an edge

Recoloring some hereditary graph classes

The reconfiguration graph of the $k$-colorings, denoted $R_k(G)$, is the graph whose vertices are the $k$-colorings of $G$ and two colorings are adjacent in $R_k(G)$ if they differ in color on exactly one vertex. A graph $G$ is said to be recolorable if $R_{\ell}(G)$ is connected for all $\ell\geq \chi(G)$+1. In this paper, we study the recolorability of several graph classes restricted by forbidden induced subgraphs. We prove some properties of a vertex-minimal graph $G$ which is not recolorable. We show that every (triangle, $H$)-free graph is recolorable if and only if every (paw, $H$)-free graph is recolorable. Every graph in the class of $(2K_2,\ H)$-free graphs, where $H$ is a 4-vertex graph except $P_4$ or $P_3$+$P_1$, is recolorable if $H$ is either a triangle, paw, claw, or diamond. Furthermore, we prove that every ($P_5$, $C_5$, house, co-banner)-free graph is recolorable.Comment: 17 page

The Complexity of the List Partition Problem for Graphs

The k-partition problem is as follows: Given a graph G and a positive integer k, partition the vertices of G into at most k parts A1, A2, . . . , Ak, where it may be specified that Ai induces a stable set, a clique, or an arbitrary subgraph, and pairs Ai, Aj (i≠j) be completely nonadjacent, completely adjacent, or arbitrarily adjacent. The list k-partition problem generalizes the k-partition problem by specifying for each vertex x, a list L(x) of parts in which it is allowed to be placed. Many well-known graph problems can be formulated as list k-partition problems: e.g., 3-colorability, clique cutset, stable cutset, homogeneous set, skew partition, and 2-clique cutset. We classify, with the exception of two polynomially equivalent problems, each list 4-partition problem as either solvable in polynomial time or NP-complete. In doing so, we provide polynomial-time algorithms for many problems whose polynomial-time solvability was open, including the list 2-clique cutset problem. This also allows us to classify each list generalized 2-clique cutset problem and list generalized skew partition problem as solvable in polynomial time or NP-complete