Independent sets in the union of two Hamiltonian cycles

Motivated by a question on the maximal number of vertex disjoint Schrijver graphs in the Kneser graph, we investigate the following function, denoted by f(n, k): the maximal number of Hamiltonian cycles on an n element set, such that no two cycles share a common independent set of size more than k. We shall mainly be interested in the behavior of f(n, k) when k is a linear function of n, namely k = cn. We show a threshold phenomenon: there exists a constant e t such that for c infinity). We prove that 0.26 12 vertices is the union of two Hamiltonian cycles and alpha(G) = n/4, then V (G) can be covered by vertex-disjoint K-4 subgraphs

Packings and coverings with Hamilton cycles and on-line Ramsey theory

A major theme in modern graph theory is the exploration of maximal packings and minimal covers of graphs with subgraphs in some given family. We focus on packings and coverings with Hamilton cycles, and prove the following results in the area. • Let ε > 0, and let $G$ be a large graph on n vertices with minimum degree at least (1=2+ ε)n. We give a tight lower bound on the size of a maximal packing of $G$ with edge-disjoint Hamilton cycles. • Let $T$ be a strongly k-connected tournament. We give an almost tight lower bound on the size of a maximal packing of $T$ with edge-disjoint Hamilton cycles. • Let log $^1$$^1$$^7$ $n$/$n$≤$p$≤1-$n$$^-$$^1$$^/$$^8$. We prove that $G$$_n$$_,$$_p$ may a.a.s be covered by a set of ⌈Δ($G$$_n$$_,$$_p$)/2⌉ Hamilton cycles, which is clearly best possible. In addition, we consider some problems in on-line Ramsey theory. Let r($G$,$H$) denote the on-line Ramsey number of $G$ and $H$. We conjecture the exact values of r ($P$$_k$,$P$$_ℓ$) for all $k$≤ℓ. We prove this conjecture for $k$=2, prove it to within an additive error of 10 for $k$=3, and prove an asymptotically tight lower bound for $k$=4. We also determine r($P$$_3$,$C$$_ℓ$ exactly for all ℓ

Some results on triangle partitions

We show that there exist efficient algorithms for the triangle packing problem in colored permutation graphs, complete multipartite graphs, distance-hereditary graphs, k-modular permutation graphs and complements of k-partite graphs (when k is fixed). We show that there is an efficient algorithm for C_4-packing on bipartite permutation graphs and we show that C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite graphs that have a triangle partition

Matching Is as Easy as the Decision Problem, in the NC Model

Is matching in NC, i.e., is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in TCS for over three decades, ever since the discovery of randomized NC matching algorithms [KUW85, MVV87]. Over the last five years, the theoretical computer science community has launched a relentless attack on this question, leading to the discovery of several powerful ideas. We give what appears to be the culmination of this line of work: An NC algorithm for finding a minimum-weight perfect matching in a general graph with polynomially bounded edge weights, provided it is given an oracle for the decision problem. Consequently, for settling the main open problem, it suffices to obtain an NC algorithm for the decision problem. We believe this new fact has qualitatively changed the nature of this open problem. All known efficient matching algorithms for general graphs follow one of two approaches: given by Edmonds [Edm65] and Lov\'asz [Lov79]. Our oracle-based algorithm follows a new approach and uses many of the ideas discovered in the last five years. The difficulty of obtaining an NC perfect matching algorithm led researchers to study matching vis-a-vis clever relaxations of the class NC. In this vein, recently Goldwasser and Grossman [GG15] gave a pseudo-deterministic RNC algorithm for finding a perfect matching in a bipartite graph, i.e., an RNC algorithm with the additional requirement that on the same graph, it should return the same (i.e., unique) perfect matching for almost all choices of random bits. A corollary of our reduction is an analogous algorithm for general graphs.Comment: Appeared in ITCS 202

Partitioning edge-coloured complete graphs into monochromatic cycles and paths

A conjecture of Erd\H{o}s, Gy\'arf\'as, and Pyber says that in any edge-colouring of a complete graph with r colours, it is possible to cover all the vertices with r vertex-disjoint monochromatic cycles. So far, this conjecture has been proven only for r = 2. In this paper we show that in fact this conjecture is false for all r > 2. In contrast to this, we show that in any edge-colouring of a complete graph with three colours, it is possible to cover all the vertices with three vertex-disjoint monochromatic paths, proving a particular case of a conjecture due to Gy\'arf\'as. As an intermediate result we show that in any edge-colouring of the complete graph with the colours red and blue, it is possible to cover all the vertices with a red path, and a disjoint blue balanced complete bipartite graph.Comment: 25 pages, 3 figure

Edge-disjoint Hamilton cycles in graphs

In this paper we give an approximate answer to a question of Nash-Williams from 1970: we show that for every \alpha > 0, every sufficiently large graph on n vertices with minimum degree at least (1/2 + \alpha)n contains at least n/8 edge-disjoint Hamilton cycles. More generally, we give an asymptotically best possible answer for the number of edge-disjoint Hamilton cycles that a graph G with minimum degree \delta must have. We also prove an approximate version of another long-standing conjecture of Nash-Williams: we show that for every \alpha > 0, every (almost) regular and sufficiently large graph on n vertices with minimum degree at least $(1/2 + \alpha)n$ can be almost decomposed into edge-disjoint Hamilton cycles.Comment: Minor Revisio