60 research outputs found
Generalizations of Grillet's theorem on maximal stable sets and maximal cliques in graphs
AbstractGrillet established conditions on a partially ordered set under which each maximal antichain meets each maximal chain. Berge pointed out that Grillet's theorem can be stated in terms of graphs, made a conjecture that strengthens it, and asked a related question. We exhibit a counterexample to the conjecture and answer the question; then we prove four theorems that generalize Grillet's theorem in the spirit of Berge's proposals
Proof of the Goldberg-Seymour Conjecture on Edge-Colorings of Multigraphs
Given a multigraph , the {\em edge-coloring problem} (ECP) is to
color the edges of with the minimum number of colors so that no two
adjacent edges have the same color. This problem can be naturally formulated as
an integer program, and its linear programming relaxation is called the {\em
fractional edge-coloring problem} (FECP). In the literature, the optimal value
of ECP (resp. FECP) is called the {\em chromatic index} (resp. {\em fractional
chromatic index}) of , denoted by (resp. ). Let
be the maximum degree of and let where is the set of all edges of with
both ends in . Clearly, is
a lower bound for . As shown by Seymour, . In the 1970s Goldberg and Seymour independently conjectured
that . Over the
past four decades this conjecture, a cornerstone in modern edge-coloring, has
been a subject of extensive research, and has stimulated a significant body of
work. In this paper we present a proof of this conjecture. Our result implies
that, first, there are only two possible values for , so an analogue
to Vizing's theorem on edge-colorings of simple graphs, a fundamental result in
graph theory, holds for multigraphs; second, although it is -hard in
general to determine , we can approximate it within one of its true
value, and find it exactly in polynomial time when ;
third, every multigraph satisfies , so FECP has a
fascinating integer rounding property
On Box-Perfect Graphs
Let be a graph and let be the clique-vertex incidence matrix
of . It is well known that is perfect iff the system , is totally dual integral (TDI). In 1982,
Cameron and Edmonds proposed to call box-perfect if the system
, is box-totally dual
integral (box-TDI), and posed the problem of characterizing such graphs. In
this paper we prove the Cameron-Edmonds conjecture on box-perfectness of parity
graphs, and identify several other classes of box-perfect graphs. We also
develop a general and powerful method for establishing box-perfectness
Coloring graphs with no odd-K4
AbstractThe purpose of this note is to present a polynomial-time algorithm which, given an arbitrary graph G as its input, finds either a proper 3-coloring of G or an odd-K4 that is a subgraph of G in time O(mn), where m and n stand for the number of edges and the number of vertices of G, respectively
Ranking tournaments with no errors I: Structural description
In this series of two papers we examine the classical problem of ranking a set of players on the basis of a set of pairwise comparisons arising from a sports tournament, with the objective of minimizing the total number of upsets, where an upset occurs if a higher ranked player was actually defeated by a lower ranked player. This problem can be rephrased as the so-called minimum feedback arc set problem on tournaments, which arises in a rich variety of applications and has been a subject of extensive research. In this series we study this NP-hard problem using structure-driven and linear programming approaches. Let T=(V,A) be a tournament with a nonnegative integral weight w(e) on each arc e. A subset F of arcs is called a feedback arc set if T\F contains no cycles (directed). A collection C of cycles (with repetition allowed) is called a cycle packing if each arc e is used at most w(e) times by members of C. We call T cycle Mengerian (CM) if, for every nonnegative integral function w defined on A, the minimum total weight of a feedback arc set is equal to the maximum size of a cycle packing. The purpose of these two papers is to show that a tournament is CM iff it contains none of four Möbius ladders as a subgraph; such a tournament is referred to as Möbius-free. In this first paper we present a structural description of all Möbius-free tournaments, which relies heavily on a chain theorem concerning internally 2-strong tournaments
Realizing Degree Sequences with Graphs Having Nowhere-Zero 3-Flows
The following open problem was proposed by Archdeacon: Characterize all graphical sequences π such that some realization of π admits a nowhere-zero 3-flow. The purpose of this paper is to resolve this problem and present a complete characterization: A graphical sequence π = (d1, d2, ., dn) with minimum degree at least two has a realization that admits a nowhere-zero 3-flow if and only if π ≠ (34, 2), (k, 3k), (k2, 3k―1), where k is an odd integer
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