Dirac's theorem for random regular graphs

We prove a `resilience' version of Dirac's theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $\varepsilon>0$, a.a.s. the following holds: let $G'$ be any subgraph of the random $n$-vertex $d$-regular graph $G_{n,d}$ with minimum degree at least $(1/2+\varepsilon)d$. Then $G'$ is Hamiltonian. This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly, the condition that $d$ is large cannot be omitted, and secondly, the minimum degree bound cannot be improved.Comment: Final accepted version, to appear in Combinatorics, Probability & Computin

Suboptimality of local algorithms for a class of max-cut problems

We show that in random K -uniform hypergraphs of constant average degree, for even K ≥ 4 , local algorithms defined as factors of i.i.d. can not find nearly maximal cuts, when the average degree is sufficiently large. These algorithms have been used frequently to obtain lower bounds for the max-cut problem on random graphs, but it was not known whether they could be successful in finding nearly maximal cuts. This result follows from the fact that the overlap of any two nearly maximal cuts in such hypergraphs does not take values in a certain nontrivial interval—a phenomenon referred to as the overlap gap property—which is proved by comparing diluted models with large average degree with appropriate fully connected spin glass models and showing the overlap gap property in the latter setting

The complexity of the L(p,q)-labeling problem for bipartite planar graphs of small degree

AbstractGiven a simple graph G, by an L(p,q)-labeling of G we mean a function c that assigns nonnegative integers to its vertices in such a way that if two vertices u, v are adjacent then |c(u)−c(v)|≥p, and if they are at distance 2 then |c(u)−c(v)|≥q. The L(p,q)-labeling problem can be defined as follows: given a graph G and integer t, determine whether there exists an L(p,q)-labeling c of G such that c(V)⊆{0,1,…,t}. In the paper we show that the problem is NP-complete even when restricted to bipartite planar graphs of small maximum degree and for relatively small values of t. More precisely, we prove that: (1)if p<3q then the problem is NP-complete for bipartite planar graphs of maximum degree Δ≤3 and t=p+max{2q,p};(2)if p=3q then the problem is NP-complete for bipartite planar graphs of maximum degree Δ≤4 and t=6q;(3)if p>3q then the problem is NP-complete for bipartite planar graphs of maximum degree Δ≤4 and t=p+5q.In particular, these results imply that the L(2,1)-labeling problem in planar graphs is NP-complete for t=4, and that the L(p,q)-labeling problem in graphs of maximum degree Δ≤4 is NP-complete for all values of p and q, thus answering two well-known open questions

On improving matchings in trees, via bounded-length augmentations

Due to a classical result of Berge, it is known that a matching of any graph can be turned into a maximum matching by repeatedly augmenting alternating paths whose ends are not covered. In a recent work, Nisse, Salch and Weber considered the influence, on this process, of augmenting paths with length at most k only. Given a graph G, an initial matching M ⊆ E(G) and an odd integer k, the problem is to find a longest sequence of augmenting paths of length at most k that can be augmented sequentially from M. They proved that, when only paths of length at most k = 3 can be augmented, computing such a longest sequence can be done in polynomial time for any graph, while the same problem for any k ≥ 5 is NP-hard. Although the latter result remains true for bipartite graphs, the status of the complexity of the same problem for trees is not known. This work is dedicated to the complexity of this problem for trees. On the positive side, we first show that it can be solved in polynomial time for more classes of trees, namely bounded-degree trees (via a dynamic programming approach), caterpillars and trees where the nodes with degree at least 3 are sufficiently far apart. On the negative side, we show that, when only paths of length exactly k can be augmented, the problem becomes NP-hard already for k = 3, in the class of planar bipartite graphs with maximum degree 3 and arbitrary large girth. We also show that the latter problem is NP-hard in trees when k is part of the input

On improving matchings in trees, via bounded-length augmentations

International audienceDue to a classical result of Berge, it is known that a matching of any graph can be turned into a maximum matching by repeatedly augmenting alternating paths whose ends are not covered. In a recent work, Nisse, Salch and Weber considered the influence, on this process, of augmenting paths with length at most k only. Given a graph G, an initial matching M ⊆ E(G) and an odd integer k, the problem is to find a longest sequence of augmenting paths of length at most k that can be augmented sequentially from M. They proved that, when only paths of length at most k = 3 can be augmented, computing such a longest sequence can be done in polynomial time for any graph, while the same problem for any k ≥ 5 is NP-hard. Although the latter result remains true for bipartite graphs, the status of the complexity of the same problem for trees is not known. This work is dedicated to the complexity of this problem for trees. On the positive side, we first show that it can be solved in polynomial time for more classes of trees, namely bounded-degree trees (via a dynamic programming approach), caterpillars and trees where the nodes with degree at least 3 are sufficiently far apart. On the negative side, we show that, when only paths of length exactly k can be augmented, the problem becomes NP-hard already for k = 3, in the class of planar bipartite graphs with maximum degree 3 and arbitrary large girth. We also show that the latter problem is NP-hard in trees when k is part of the input