5/4-Approximation of Minimum 2-Edge-Connected Spanning Subgraph

We provide a $5/4$-approximation algorithm for the minimum 2-edge-connected spanning subgraph problem. This improves upon the previous best ratio of $4/3$. The algorithm is based on applying local improvement steps on a starting solution provided by a standard ear decomposition together with the idea of running several iterations on residual graphs by excluding certain edges that do not belong to an optimum solution. The latter idea is a novel one, which allows us to bypass $3$-ears with no loss in approximation ratio, the bottleneck for obtaining a performance guarantee below $3/2$. Our algorithm also implies a simpler $7/4$-approximation algorithm for the matching augmentation problem, which was recently treated.Comment: The modification of 5-ears, which was both erroneous and unnecessary, is omitte

A general formula for the index of depth stability of edge ideals

By a classical result of Brodmann, the function $\operatorname{depth} R/I^t$ is asymptotically a constant, i.e. there is a number $s$ such that $\operatorname{depth} R/I^t = \operatorname{depth} R/I^s$ for $t > s$. One calls the smallest number $s$ with this property the index of depth stability of $I$ and denotes it by $\operatorname{dstab}(I)$. This invariant remains mysterious til now. The main result of this paper gives an explicit formula for $\operatorname{dstab}(I)$ when $I$ is an arbitrary ideal generated by squarefree monomials of degree 2. That is the first general case where one can characterize $\operatorname{dstab}(I)$ explicitly. The formula expresses $\operatorname{dstab}(I)$ in terms of the associated graph. The proof involves new techniques which relate different topics such as simplicial complexes, systems of linear inequalities, graph parallelizations, and ear decompositions. It provides an effective method for the study of powers of edge ideals.Comment: 23 pages, 4 figure

Maximality of the cycle code of a graph

AbstractThe cycle code of a graph is the binary linear span of the characteristic vectors of circuits. We characterize the graphs whose cycle codes are maximal for the packing problem, based on characterizing the graphs whose girth is at least 12(n−c) +1 where n and c are the numbers of vertices and connected components

Decreasing behavior of the depth functions of edge ideals

Let $I$ be the edge ideal of a connected non-bipartite graph and $R$ the base polynomial ring. Then $\operatorname{depth} R/I \ge 1$ and $\operatorname{depth} R/I^t = 0$ for $t \gg 1$. We give combinatorial conditions for $\operatorname{depth} R/I^t = 1$ for some $t$ in between and show that the depth function is non-increasing thereafter. Especially, the depth function quickly decreases to 0 after reaching 1. We show that if $\operatorname{depth} R/I = 1$ then $\operatorname{depth} R/I^2 = 0$ and if $\operatorname{depth} R/I^2 = 1$ then $\operatorname{depth} R/I^5 = 0$. Other similar results suggest that if $\operatorname{depth} R/I^t = 1$ then $\operatorname{depth} R/I^{t+3} = 0$. This a surprising phenomenon because the depth of a power can determine a smaller depth of another power. Furthermore, we are able to give a simple combinatorial criterion for $\operatorname{depth} R/I^{(t)} = 1$ for $t \gg 1$ and show that the condition $\operatorname{depth} R/I^{(t)} = 1$ is persistent, where $I^{(t)}$ denotes the $t$-th symbolic powers of $I$.Comment: 15 pages, 3 figure