Optimal Bounds for the kk-cut Problem

In the $k$-cut problem, we want to find the smallest set of edges whose deletion breaks a given (multi)graph into $k$ connected components. Algorithms of Karger & Stein and Thorup showed how to find such a minimum $k$-cut in time approximately $O(n^{2k})$. The best lower bounds come from conjectures about the solvability of the $k$-clique problem, and show that solving $k$-cut is likely to require time $\Omega(n^k)$. Recent results of Gupta, Lee & Li have given special-purpose algorithms that solve the problem in time $n^{1.98k + O(1)}$, and ones that have better performance for special classes of graphs (e.g., for small integer weights). In this work, we resolve the problem for general graphs, by showing that the Contraction Algorithm of Karger outputs any fixed $k$-cut of weight $\alpha \lambda_k$ with probability $\Omega_k(n^{-\alpha k})$, where $\lambda_k$ denotes the minimum $k$-cut size. This also gives an extremal bound of $O_k(n^k)$ on the number of minimum $k$-cuts and an algorithm to compute a minimum $k$-cut in similar runtime. Both are tight up to lower-order factors, with the algorithmic lower bound assuming hardness of max-weight $k$-clique. The first main ingredient in our result is a fine-grained analysis of how the graph shrinks -- and how the average degree evolves -- in the Karger process. The second ingredient is an extremal bound on the number of cuts of size less than $2 \lambda_k/k$, using the Sunflower lemma.Comment: Final version of arXiv:1911.09165 with new and more general proof

On matchings and factors of graphs /

In Section 1, we recall the historical sketch of matching and factor theory of graphs, and also introduce some necessary definitions and notation. In Section 2, we present a sufficient condition for the existence of a (g, f)-factor in graphs with the odd-cycle property, which is simpler than that of Lovasz\u27s (g, f)-Factor Theorem. From this, we derive some further results, and we show that (a) every r-regular graph G with the odd-cycle property has a k-factor, where 0 ≤ k ≤ r and k|V(G)| ≡ 0 (mod 2), (b) every graph G with the strong odd-cycle property with k|V(G)|≡ 0 (mod 2) is k-factorable if and only if G is a km-regular graph for some m ≥ 1, and (c) every regular graph of even order with the strong odd-cycle property is of the second class (i.e. the edge chromatic number is Δ). Chvátal [26] presented the following two conjectures that (1) a graph G has a 2-factor if tough(G) ≥ 3/2, and (2) a graph G has a k-factor if k|V(G)| ≡ 0 (mod 2) and tough(G) ≥ k. Enomoto et.al. [32] proved the second conjecture. They also proved the sharpness of the bound on tough(G) that guarantees the existence of a k-factor. This implies that the first conjecture is false. In Section 3, we show that the result of the second conjecture can be improved in some sense, and the first conjecture is also true if the graph considered has the odd-cycle property. Anderson [3] stated that a graph G of even order has a 1-factor if bind(G) ≥ 4/3, and Katerinis and Woodall [48] proved that a graph G of order n has a k-factor if bind(G) ˃ (2k -I)(n - 1)/(k(n - 2) + 3), where k ≥ 2, n ≥ 4k - 6 and kn ≡ 0 (mod 2). In Section 4, we shall present some similar conditions for the existence of [a, b]-factors. In Section 5, we study the existence of [a, b]-parity-factors in a graph, among which we extend some known theorems from 1-factors to {1, 3, ... , 2n - 1}-factors, or from k-factors to [a, b]-parity-factors. Also, extending Petersen\u27s 2-Factorization Theorem, we proved that a graph is [2a, 2b]-even-factorable if and only if it is a [2na, 2nb]-even-graph for some n ≥ 1. Plummer showed that (a) (in [58]) every graph G of even order is k-extendable if tough(G) ˃ k, and (b) (in [59]) every (2k+1)-connected graph G is k-extendable if G is K1,3-free, respectively. In Section 6, we give a counterpart of the former in terms of binding number, and extend the latter from K1,3-free graphs to K1,n-free graphs. Furthermore, we present a result toward the problem, posed by Saito [61] and Plummer [60], of characterizing the graphs that are maximal k-extendable

Ramsey-nice families of graphs

For a finite family $\mathcal{F}$ of fixed graphs let $R_k(\mathcal{F})$ be the smallest integer $n$ for which every $k$-coloring of the edges of the complete graph $K_n$ yields a monochromatic copy of some $F\in\mathcal{F}$. We say that $\mathcal{F}$ is $k$-nice if for every graph $G$ with $\chi(G)=R_k(\mathcal{F})$ and for every $k$-coloring of $E(G)$ there exists a monochromatic copy of some $F\in\mathcal{F}$. It is easy to see that if $\mathcal{F}$ contains no forest, then it is not $k$-nice for any $k$. It seems plausible to conjecture that a (weak) converse holds, namely, for any finite family of graphs $\mathcal{F}$ that contains at least one forest, and for all $k\geq k_0(\mathcal{F})$ (or at least for infinitely many values of $k$), $\mathcal{F}$ is $k$-nice. We prove several (modest) results in support of this conjecture, showing, in particular, that it holds for each of the three families consisting of two connected graphs with 3 edges each and observing that it holds for any family $\mathcal{F}$ containing a forest with at most 2 edges. We also study some related problems and disprove a conjecture by Aharoni, Charbit and Howard regarding the size of matchings in regular 3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure

Powers of Hamilton cycles in pseudorandom graphs

We study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph $G$ is $(\varepsilon,p,k,\ell)$-pseudorandom if for all disjoint $X$ and $Y\subset V(G)$ with $|X|\ge\varepsilon p^kn$ and $|Y|\ge\varepsilon p^\ell n$ we have $e(X,Y)=(1\pm\varepsilon)p|X||Y|$. We prove that for all $\beta>0$ there is an $\varepsilon>0$ such that an $(\varepsilon,p,1,2)$-pseudorandom graph on $n$ vertices with minimum degree at least $\beta pn$ contains the square of a Hamilton cycle. In particular, this implies that $(n,d,\lambda)$-graphs with $\lambda\ll d^{5/2 }n^{-3/2}$ contain the square of a Hamilton cycle, and thus a triangle factor if $n$ is a multiple of $3$. This improves on a result of Krivelevich, Sudakov and Szab\'o [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), no. 3, 403--426]. We also extend our result to higher powers of Hamilton cycles and establish corresponding counting versions.Comment: 30 pages, 1 figur

Measures of edge-uncolorability

The resistance $r(G)$ of a graph $G$ is the minimum number of edges that have to be removed from $G$ to obtain a graph which is $\Delta(G)$-edge-colorable. The paper relates the resistance to other parameters that measure how far is a graph from being $\Delta$-edge-colorable. The first part considers regular graphs and the relation of the resistance to structural properties in terms of 2-factors. The second part studies general (multi-) graphs $G$. Let $r_v(G)$ be the minimum number of vertices that have to be removed from $G$ to obtain a class 1 graph. We show that $\frac{r(G)}{r_v(G)} \leq \lfloor \frac{\Delta(G)}{2} \rfloor$, and that this bound is best possible.Comment: 9 page

Petersen cores and the oddness of cubic graphs

Let $G$ be a bridgeless cubic graph. Consider a list of $k$ 1-factors of $G$. Let $E_i$ be the set of edges contained in precisely $i$ members of the $k$ 1-factors. Let $\mu_k(G)$ be the smallest $|E_0|$ over all lists of $k$ 1-factors of $G$. If $G$ is not 3-edge-colorable, then $\mu_3(G) \geq 3$. In [E. Steffen, 1-factor and cycle covers of cubic graphs, J. Graph Theory 78(3) (2015) 195-206] it is shown that if $\mu_3(G) \not = 0$, then $2 \mu_3(G)$ is an upper bound for the girth of $G$. We show that $\mu_3(G)$ bounds the oddness $\omega(G)$ of $G$ as well. We prove that $\omega(G)\leq \frac{2}{3}\mu_3(G)$. If $\mu_3(G) = \frac{2}{3} \mu_3(G)$, then every $\mu_3(G)$-core has a very specific structure. We call these cores Petersen cores. We show that for any given oddness there is a cyclically 4-edge-connected cubic graph $G$ with $\omega(G) = \frac{2}{3}\mu_3(G)$. On the other hand, the difference between $\omega(G)$ and $\frac{2}{3}\mu_3(G)$ can be arbitrarily big. This is true even if we additionally fix the oddness. Furthermore, for every integer $k\geq 3$, there exists a bridgeless cubic graph $G$ such that $\mu_3(G)=k$.Comment: 13 pages, 9 figure