Fractional total colourings of graphs of high girth

Reed conjectured that for every epsilon>0 and Delta there exists g such that the fractional total chromatic number of a graph with maximum degree Delta and girth at least g is at most Delta+1+epsilon. We prove the conjecture for Delta=3 and for even Delta>=4 in the following stronger form: For each of these values of Delta, there exists g such that the fractional total chromatic number of any graph with maximum degree Delta and girth at least g is equal to Delta+1

Round and Bipartize for Vertex Cover Approximation

The vertex cover problem is a fundamental and widely studied combinatorial optimization problem. It is known that its standard linear programming relaxation is integral for bipartite graphs and half-integral for general graphs. As a consequence, the natural rounding algorithm based on this relaxation computes an optimal solution for bipartite graphs and a 2-approximation for general graphs. This raises the question of whether one can interpolate the rounding curve of the standard linear programming relaxation in a beyond the worst-case manner, depending on how close the graph is to being bipartite. In this paper, we consider a round-and-bipartize algorithm that exploits the knowledge of an induced bipartite subgraph to attain improved approximation ratios. Equivalently, we suppose that we work with a pair (?, S), consisting of a graph with an odd cycle transversal. If S is a stable set, we prove a tight approximation ratio of 1 + 1/?, where 2? -1 denotes the odd girth (i.e., length of the shortest odd cycle) of the contracted graph ?? : = ?/S and satisfies ? ? [2,?], with ? = ? corresponding to the bipartite case. If S is an arbitrary set, we prove a tight approximation ratio of (1+1/?) (1 - ?) + 2 ?, where ? ? [0,1] is a natural parameter measuring the quality of the set S. The technique used to prove tight improved approximation ratios relies on a structural analysis of the contracted graph ??, in combination with an understanding of the weight space where the fully half-integral solution is optimal. Tightness is shown by constructing classes of weight functions matching the obtained upper bounds. As a byproduct of the structural analysis, we also obtain improved tight bounds on the integrality gap and the fractional chromatic number of 3-colorable graphs. We also discuss algorithmic applications in order to find good odd cycle transversals, connecting to the MinUncut and Colouring problems. Finally, we show that our analysis is optimal in the following sense: the worst case bounds for ? and ?, which are ? = 2 and ? = 1 - 4/n, recover the integrality gap of 2 - 2/n of the standard linear programming relaxation, where n is the number of vertices of the graph

The Strong Perfect Graph Conjecture: 40 years of Attempts, and its Resolution

International audienceThe Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The first of these three approaches yielded the first (and to date only) proof of the SPGC; the other two remain promising to consider in attempting an alternative proof. This paper is an unbalanced survey of the attempts to solve the SPGC; unbalanced, because (1) we devote a signicant part of it to the 'primitive graphs and structural faults' paradigm which led to the Strong Perfect Graph Theorem (SPGT); (2) we briefly present the other "direct" attempts, that is, the ones for which results exist showing one (possible) way to the proof; (3) we ignore entirely the "indirect" approaches whose aim was to get more information about the properties and structure of perfect graphs, without a direct impact on the SPGC. Our aim in this paper is to trace the path that led to the proof of the SPGT as completely as possible. Of course, this implies large overlaps with the recent book on perfect graphs [J.L. Ramirez-Alfonsin and B.A. Reed, eds., Perfect Graphs (Wiley & Sons, 2001).], but it also implies a deeper analysis (with additional results) and another viewpoint on the topic