Degree Sums, k-Factors and Hamilton Cycles in Graphs

https://digitalcommons.memphis.edu/speccoll-faudreerj/1189/thumbnail.jp

Long cycles in graphs containing a 2-factor with many odd components

We prove a result on the length of a longest cycle in a graph on n vertices that contains a 2-factor and satisfies d(u)+d(c)+d(w)n+2 for every tiple u, v, w of independent vertices. As a corollary we obtain the follwing improvement of a conjectre of Häggkvist (1992): Let G be a 2-connected graph on n vertices where every pair of nonadjacent vertices has degree sum at least n-k and assume G has a 2-factor with at least k+1 odd components. Then G is hamiltonian

Packing Hamilton Cycles Online

It is known that w.h.p. the hitting time $\tau_{2\sigma}$ for the random graph process to have minimum degree $2\sigma$ coincides with the hitting time for $\sigma$ edge disjoint Hamilton cycles. In this paper we prove an online version of this property. We show that, for a fixed integer $\sigma\geq 2$, if random edges of $K_n$ are presented one by one then w.h.p. it is possible to color the edges online with $\sigma$ colors so that at time $\tau_{2\sigma}$, each color class is Hamiltonian.Comment: Minor change

Cycle factors and renewal theory

For which values of $k$ does a uniformly chosen $3$-regular graph $G$ on $n$ vertices typically contain $n/k$ vertex-disjoint $k$-cycles (a $k$-cycle factor)? To date, this has been answered for $k=n$ and for $k \ll \log n$; the former, the Hamiltonicity problem, was finally answered in the affirmative by Robinson and Wormald in 1992, while the answer in the latter case is negative since with high probability most vertices do not lie on $k$-cycles. Here we settle the problem completely: the threshold for a $k$-cycle factor in $G$ as above is $\kappa_0 \log_2 n$ with $\kappa_0=[1-\frac12\log_2 3]^{-1}\approx 4.82$. Precisely, we prove a 2-point concentration result: if $k \geq \kappa_0 \log_2(2n/e)$ divides $n$ then $G$ contains a $k$-cycle factor w.h.p., whereas if $k<\kappa_0\log_2(2n/e)-\frac{\log^2 n}n$ then w.h.p. it does not. As a byproduct, we confirm the "Comb Conjecture," an old problem concerning the embedding of certain spanning trees in the random graph $G(n,p)$. The proof follows the small subgraph conditioning framework, but the associated second moment analysis here is far more delicate than in any earlier use of this method and involves several novel features, among them a sharp estimate for tail probabilities in renewal processes without replacement which may be of independent interest.Comment: 45 page

P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches

While the P vs NP problem is mainly approached form the point of view of discrete mathematics, this paper proposes reformulations into the field of abstract algebra, geometry, fourier analysis and of continuous global optimization - which advanced tools might bring new perspectives and approaches for this question. The first one is equivalence of satisfaction of 3-SAT problem with the question of reaching zero of a nonnegative degree 4 multivariate polynomial (sum of squares), what could be tested from the perspective of algebra by using discriminant. It could be also approached as a continuous global optimization problem inside $[0,1]^n$, for example in physical realizations like adiabatic quantum computers. However, the number of local minima usually grows exponentially. Reducing to degree 2 polynomial plus constraints of being in $\{0,1\}^n$, we get geometric formulations as the question if plane or sphere intersects with $\{0,1\}^n$. There will be also presented some non-standard perspectives for the Subset-Sum, like through convergence of a series, or zeroing of $\int_0^{2\pi} \prod_i \cos(\varphi k_i) d\varphi$ fourier-type integral for some natural $k_i$. The last discussed approach is using anti-commuting Grassmann numbers $\theta_i$, making $(A \cdot \textrm{diag}(\theta_i))^n$ nonzero only if $A$ has a Hamilton cycle. Hence, the P$\ne$NP assumption implies exponential growth of matrix representation of Grassmann numbers. There will be also discussed a looking promising algebraic/geometric approach to the graph isomorphism problem -- tested to successfully distinguish strongly regular graphs with up to 29 vertices.Comment: 19 pages, 8 figure

On some intriguing problems in Hamiltonian graph theory -- A survey

We survey results and open problems in Hamiltonian graph theory centred around three themes: regular graphs, $t$-tough graphs, and claw-free graphs

Hipergráfok = Hypergraphs

A projekt célkitűzéseit sikerült megvalósítani. A négy év során több mint száz kiváló eredmény született, amiből eddig 84 dolgozat jelent meg a téma legkiválóbb folyóirataiban, mint Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, stb. Számos régóta fennálló sejtést bebizonyítottunk, egész régi nyitott problémát megoldottunk hipergráfokkal kapcsolatban illetve kapcsolódó területeken. A problémák némelyike sok éve, olykor több évtizede nyitott volt. Nem egy közvetlen kutatási eredmény, de szintén bizonyos értékmérő, hogy a résztvevők egyike a Norvég Királyi Akadémia tagja lett és elnyerte a Steele díjat. | We managed to reach the goals of the project. We achieved more than one hundred excellent results, 84 of them appeared already in the most prestigious journals of the subject, like Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, etc. We proved several long standing conjectures, solved quite old open problems in the area of hypergraphs and related subjects. Some of the problems were open for many years, sometimes for decades. It is not a direct research result but kind of an evaluation too that a member of the team became a member of the Norvegian Royal Academy and won Steele Prize