Ge 2k Research Articles

Partial Degree Conditions and Cycle Coverings in Bipartite Graphs

Let k be a positive integer. Let G be a graph of order $$n\ge 3$$nź3 and W a subset of V(G) with $$|W|\ge 3k$$|W|ź3k. Wang (J Graph Theory 78:295---304, 2015) proved that if $$d(x)\ge 2n/3$$d(x)ź2n...

Partial Degree Conditions and Cycle Coverings in Bipartite Graphs

Let k be a positive integer. Let G be a graph of order $$n\ge 3$$n?3 and W a subset of V(G) with $$|W|\ge 3k$$|W|?3k. Wang (J Graph Theory 78:295---304, 2015) proved that if $$d(x)\ge 2n/3$$d(x)?2n/3 for each $$x\in W$$x?W, then G contains k vertex-disjoint cycles such that each of them contains at least three vertices of W. In this paper, we obtain an analogue result of Wang's Theorem in bipartite graph with the partial degree condition. Let $$G=(V_1,V_2;E)$$G=(V1,V2?E) be a bipartite graph with $$|V_1|=|V_2|=n$$|V1|=|V2|=n, and let W be a subset of $$V_1$$V1 with $$|W|\ge 2k$$|W|?2k, where k is a positive integer. We show that if $$d(x)+d(y)\ge n+k$$d(x)+d(y)?n+k for every pair of nonadjacent vertices $$x\in W, y\in V_2$$x?W,y?V2, then G contains k vertex-disjoint cycles such that each of them contains at least two vertices of W.

Sharpening an Ore-type version of the Corrádi–Hajnal theorem

Corradi and Hajnal (Acta Math Acad Sci Hung 14:423–439, 1963) proved that for all \(k\ge 1\) and \(n\ge 3k\), every (simple) graph G on n vertices with minimum degree \(\delta (G)\ge 2k\) contains k disjoint cycles. The degree bound is sharp. Enomoto and Wang proved the following Ore-type refinement of the Corradi–Hajnal theorem: For all \(k\ge 1\) and \(n\ge 3k\), every graph G on n vertices contains k disjoint cycles, provided that \(d(x)+d(y)\ge 4k-1\) for all distinct nonadjacent vertices x, y. Very recently, it was refined for \(k\ge 3\) and \(n\ge 3k+1\): If G is a graph on n vertices such that \(d(x)+d(y)\ge 4k-3\) for all distinct nonadjacent vertices x, y, then G has k vertex-disjoint cycles if and only if the independence number \(\alpha (G)\le n-2k\) and G is not one of two small exceptions in the case \(k=3\). But the most difficult case, \(n=3k\), was not handled. In this case, there are more exceptional graphs, the statement is more sophisticated, and some of the proofs do not work. In this paper we resolve this difficult case and obtain the full picture of extremal graphs for the Ore-type version of the Corradi–Hajnal theorem. Since any k disjoint cycles in a 3k-vertex graph G must be 3-cycles, the existence of such k cycles is equivalent to the existence of an equitable k-coloring of the complement of G. Our proof uses the language of equitable colorings, and our result can be also considered as an Ore-type version of a partial case of the Chen–Lih–Wu Conjecture on equitable colorings.

A Refinement of Theorems on Vertex-Disjoint Chorded Cycles

In 1963, Corradi and Hajnal settled a conjecture of Erd?s by proving that, for all $$k \ge 1$$k?1, any graph G with $$|G| \ge 3k$$|G|?3k and minimum degree at least 2k contains k vertex-disjoint cycles. In 2008, Finkel proved that for all $$k \ge 1$$k?1, any graph G with $$|G| \ge 4k$$|G|?4k and minimum degree at least 3k contains k vertex-disjoint chorded cycles. Finkel's result was strengthened by Chiba, Fujita, Gao, and Li in 2010, who showed, among other results, that for all $$k \ge 1$$k?1, any graph G with $$|G| \ge 4k$$|G|?4k and minimum Ore-degree at least $$6k-1$$6k-1 contains k vertex-disjoint chorded cycles. We refine this result, characterizing the graphs G with $$|G| \ge 4k$$|G|?4k and minimum Ore-degree at least $$6k-2$$6k-2 that do not have k vertex-disjoint chorded cycles.

All Ramsey Numbers for Brooms in Graphs

For $k,\ell\ge 1$, a broom $B_{k,\ell}$ is a tree on $n=k+\ell$ vertices obtained by connecting the central vertex of a star $K_{1,k}$ with an end-vertex of a path on $\ell-1$ vertices. As $B_{n-2,2}$ is a star and $B_{1,n-1}$ is a path, their Ramsey number have been determined among rarely known $R(T_n)$ of trees $T_n$ of order $n$. Erdős, Faudree, Rousseau and Schelp determined the value of $R(B_{k,\ell})$ for $\ell\ge 2k\geq2$. We shall determine all other $R(B_{k,\ell})$ in this paper, which says that, for fixed $n$, $R(B_{n-\ell,\ell})$ decreases first on $1\le\ell \le 2n/3$ from $2n-2$ or $2n-3$ to $\lceil\frac{4n}{3}\rceil-1$, and then it increases on $2n/3 < \ell\leq n$ from $\lceil\frac{4n}{3}\rceil-1$ to $\lfloor\frac{3n}{2}\rfloor -1$. Hence $R(B_{n-\ell,\ell})$ may attain the maximum and minimum values of $R(T_n)$ as $\ell$ varies.

Distances on nanotubical structures

Nanotubical graphs are obtained by wrapping a hexagonal grid into a cylinder, and then possibly closing the tube with patches. In this paper we determine the number of vertices at distance d from a particular vertex in an open (k, l) nanotubical graph. Surprisingly, this number does not depend much on the type of the nanotubical structure, but mainely on its circumference. In particular, for \(d\ge 2k\) it is \(2\,(k+l)\) for an infinite open nanotube. This result can be used as a tool for precise evaluation of distance based topological indices for nanotubical structures. The presented results imply an interesting conclusion that these indices do not distinguish the type of the nanotubes very well.

Vertex-Disjoint Cycles Containing Specified Vertices in a Bipartite Graph

The theory of vertex-disjoint cycles of a graph is the generalization of the well-known Hamiltonian cycle theory. In this paper, we prove the following result. Let $$G = (V_{1}, V_{2}; E$$G=(V1,V2źE) be a bipartite graph with $$|V_{1}|= |V_{2}|= n$$|V1|=|V2|=n such that $$n\ge 2k + 1$$nź2k+1, where k$$\ge $$ź 1 is an integer. If $$\sigma _{1,1}(G) \ge n + k$$ź1,1(G)źn+k, then for any k distinct vertices $$v_{1}, v_{2}, \ldots , v_{k}$$v1,v2,ź,vk of G, G contains $$k - 1$$k-1 quadrilaterals $$C_{1}, C_{2}, \ldots , C_{k-1}$$C1,C2,ź,Ck-1 and a path $$P_{k}$$Pk of order 2t, where $$t = n - 2(k - 1)$$t=n-2(k-1), such that all of them are vertex-disjoint and $$v_{i} \in V(C_{i})$$viźV(Ci) for each $$i \in \{1, 2, \ldots , k - 1\}, v_{k} \in V(P_{k})$$iź{1,2,ź,k-1},vkźV(Pk). Using this result we also prove that G contains k vertex-disjoint cycles $$C_{1}, C_{2}, \ldots , C_{k}$$C1,C2,ź,Ck such that $$v_{i} \in V(C_{i})$$viźV(Ci) for each $$i \in \{1, 2, \ldots , k\}$$iź{1,2,ź,k} and there are $$k - 1$$k-1 quadrilaterals in $$\{C_{1}, C_{2}, \ldots , C_{k}\}$${C1,C2,ź,Ck}. Moreover, the degree condition is sharp.

Two-dimensional direction finding of coherent signals with a linear array of vector hydrophones

In this paper, we propose a parallel factor (PARAFAC) analysis based two dimensional direction finding algorithm for coherent signals using a uniformly linear array of vector hydrophones. By forming a PARAFAC model using spatial signature of vector hydrophone array, the new algorithm requires neither spatial smoothing nor vector-field smoothing to decorrelate the signal coherency. We also establish that the azimuth-elevation angles of K coherent signals can be uniquely determined by PARAFAC analysis, as long as the number of hydrophones $$L \ge 2K - 1$$Lź2K-1. In addition, because the vector hydrophone array manifold contains no phase information, this new algorithm can offer high azimuth-elevation estimation accuracy by setting vector hydrophones to space much farther apart than a half-wavelength. Simulation results are finally presented to verify the efficacy of the proposed algorithm.

A Generalization of Multiple Choice Balls-into-Bins: Tight Bounds

This paper investigates a general version of the multiple choice model called the (k, d)-choice process in which n balls are assigned to n bins. In the process, $$kn$$m>n balls are placed into n bins is also presented for the case $$d \ge 2k$$d?2k. Potential applications are discussed such as distributed storage as well as parallel job scheduling in a cluster.

AkF¯chiral gauge theories

We study asymptotically free chiral gauge theories with an SU($N$) gauge group and chiral fermions transforming according to the antisymmetric rank-$k$ tensor representation, $A_k \equiv [k]_N$, and the requisite number, $n_{\bar F}$, of copies of fermions in the conjugate fundamental representation, $\bar F \equiv \overline{[1]}_N$, to render the theories anomaly-free. We denote these as $A_k \, \bar F$ theories. We take $N \ge 2k+1$ so that $n_{\bar F} \ge 1$. The $A_2 \, \bar F$ theories form an infinite family with $N \ge 5$, but we show that the $A_3 \, \bar F$ and $A_4 \,\bar F$ theories are only asymptotically free for $N$ in the respective ranges $7 \le N \le 17$ and $9 \le N \le 11$, and that there are no asymptotically free $A_k \, \bar F$ theories with $k \ge 5$. We investigate the types of ultraviolet to infrared evolution for these $A_k \, \bar F$ theories and find that, depending on $k$ and $N$, they may lead to a non-Abelian Coulomb phase, or may involve confinement with massless gauge-singlet composite fermions, bilinear fermion condensation with dynamical gauge and global symmetry breaking, or formation of multifermion condensates that preserve the gauge symmetry. We also show that there are no asymptotically free, anomaly-free SU($N$) $S_k \, \bar F$ chiral gauge theories with $k \ge 3$, where $S_k$ denotes the rank-$k$ symmetric representation.

The (2k-1)-connected multigraphs with at most k-1 disjoint cycles

In 1963, Corr\'adi and Hajnal proved that for all $k \ge 1$ and $n \ge 3k$, every (simple) graph on n vertices with minimum degree at least 2k contains k disjoint cycles. The same year, Dirac described the 3-connected multigraphs not containing two disjoint cycles and asked the more general question: Which (2k-1)-connected multigraphs do not contain k disjoint cycles? Recently, the authors characterized the simple graphs G with minimum degree $\delta(G) \ge 2k-1$ that do not contain k disjoint cycles. We use this result to answer Dirac's question in full.

Most Probably Intersecting Hypergraphs

The celebrated Erdős-Ko-Rado theorem shows that for $n \ge 2k$ the largest intersecting $k$-uniform set family on $[n]$ has size $\binom{n-1}{k-1}$. It is natural to ask how far from intersecting larger set families must be. Katona, Katona and Katona introduced the notion of most probably intersecting families, which maximise the probability of random subfamilies being intersecting.We consider the most probably intersecting problem for $k$-uniform set families. We provide a rough structural characterisation of the most probably intersecting families and, for families of particular sizes, show that the initial segment of the lexicographic order is optimal.

WARING'S PROBLEM WITH SHIFTS

Let $\mu_1, \ldots, \mu_s$ be real numbers, with $\mu_1$ irrational. We investigate sums of shifted $k$th powers $\mathfrak{F}(x_1, \ldots, x_s) = (x_1 - \mu_1)^k + \ldots + (x_s - \mu_s)^k$. For $k \ge 4$, we bound the number of variables needed to ensure that if $\eta$ is real and $\tau > 0$ is sufficiently large then there exist integers $x_1 > \mu_1, \ldots, x_s > \mu_s$ such that $|\mathfrak{F}(\mathbf{x}) - \tau| < \eta$. This is a real analogue to Waring's problem. When $s \ge 2k^2-2k+3$, we provide an asymptotic formula. We prove similar results for sums of general univariate degree $k$ polynomials.

Binary Codes and Partial Permutation Decoding Sets from the Johnson Graphs

For $$k \ge 1$$k?1, $$n \ge 2k$$n?2k, the Johnson graph denoted by $$J(n,k),$$J(n,k), is the graph with vertex-set the set of all $$k$$k-subsets of $$\Omega = \{1, 2, \ldots , n\}$$Ω={1,2,?,n}, and any two vertices $$u$$u and $$v$$v are adjacent if and only if $$|u \cap v| = k-1$$|u?v|=k-1. In this paper the binary codes and their duals generated by an adjacency matrix of $$J(n,k)$$J(n,k) are described. The automorphism groups of the codes are determined, and by identifying suitable information sets, 3-PD-sets are determined for the code when $$k$$k is even.

Decomposition of Complete Bipartite Digraphs and Complete Digraphs into Directed Paths and Directed Cycles of Fixed Even Length

In this paper, we give some necessary and sufficient conditions for decomposing the complete bipartite digraphs $${\mathcal D}K_{m, n}$$DKm,n and complete digraphs $${\mathcal D}K_n$$DKn into directed paths $${\mathop {P}\limits ^{\rightarrow }}_{k+1}$$P?k+1 and directed cycles $${\mathop {C}\limits ^{\rightarrow }}_{k}$$C?k with $$k$$k arcs each. In particular, we prove that: (1) For any nonnegative integers $$p$$p and $$q$$q; and any positive integers $$m$$m, $$n$$n, and $$k$$k with $$m\ge k$$m?k and $$n\ge k$$n?k; a decomposition of $${\mathcal D} K_{m, n}$$DKm,n into $$p$$p copies of $${\mathop {P}\limits ^{\rightarrow }}_{k+1}$$P?k+1 and $$q$$q copies of $${\mathop {C}\limits ^{\rightarrow }}_{k}$$C?k exists if and only if $$k(p+q)=2mn$$k(p+q)=2mn, $$p\ne 1$$p?1, $$(m, n, k, p)\ne (2, 2, 2, 3)$$(m,n,k,p)?(2,2,2,3), and $$k$$k is even when $$q>0$$q>0. (2) For any nonnegative integers $$p$$p and $$q$$q and any positive integers $$n$$n and $$k$$k with $$k$$k even and $$n\ge 2k$$n?2k, a decomposition of $${\mathcal D} K_{n}$$DKn into $$p$$p copies of $${\mathop {P}\limits ^{\rightarrow }}_{k+1}$$P?k+1 and $$q$$q copies of $${\mathop {C}\limits ^{\rightarrow }}_{k}$$C?k exists if and only if $$k(p+q)=n(n-1)$$k(p+q)=n(n-1) and $$p\ne 1$$p?1. We also give necessary and sufficient conditions for such decompositions to exist when $$k=2$$k=2 or $$4$$4.

Coverings by Few Monochromatic Pieces: A Transition Between Two Ramsey Problems

The typical problem in (generalized) Ramsey theory is to find the order of the largest monochromatic member of a family \({\mathcal{F}}\) (for example matchings, paths, cycles, connected subgraphs) that must be present in any edge coloring of a complete graph K n with t colors. Another area is to find the minimum number of monochromatic members of \({\mathcal{F}}\) that partition or cover the vertex set of every edge colored complete graph. Here we propose a problem that connects these areas: for a fixed positive integers s ≤ t, at least how many vertices can be covered by the vertices of no more than s monochromatic members of \({\mathcal{F}}\) in every edge coloring of K n with t colors. Several problems and conjectures are presented, among them a possible extension of a well-known result of Cockayne and Lorimer on monochromatic matchings for which we prove an initial step: every t-coloring of K n contains a (t − 1)-colored matching of size k provided that $$n\ge 2k +\left\lfloor{k-1\over 2^{t-1}-1} \right\rfloor.$$

Analytic varieties with finite volume amoebas are algebraic

Abstract In this paper, we study the amoeba volume of a given k-dimensional generic analytic variety V of the complex algebraic torus ( ℂ * ) n $(\mathbb {C}^*)^n$ . When n ≥ 2 k $n\ge 2k$ , we show that V is algebraic if and only if the volume of its amoeba is finite. Moreover, in this case, we establish a comparison theorem for the volume of the amoeba and the coamoeba. Examples and applications to the k-dimensional affine linear spaces are also given.

Birationally rigid Fano complete intersections. II

Abstract. We prove that a generic (in the sense of Zariski topology) Fano complete intersection V of the type ( d 1 , ⋯ , d k ) $(d_1,\dots ,d_k)$ in ℙ M + k ${\mathbb {P}}^{M+k}$ , where d 1 + ⋯ + d k = M + k $d_1+\dots +d_k=M+k$ , is birationally superrigid if M ≥ 7 $M\ge 7$ , M ≥ k + 3 $M\ge k+3$ and max { d i } ≥ 4 $\operatorname{max} \lbrace d_i\rbrace \ge 4$ . In particular, on the variety V there is exactly one structure of a Mori fibre space (or a rationally connected fibre space), the groups of birational and biregular self-maps coincide, Bir V = Aut V $\operatorname{Bir} V= \operatorname{Aut} V$ , and the variety V is non-rational. This fact covers a considerably larger range of complete intersections than our result of [J. reine angew. Math 541 (2001), 55–79], which required the condition M ≥ 2 k + 1 $M\ge 2k+1$ .

A Rainbow $k$-Matching in the Complete Graph with $r$ Colors

An $r$-edge-coloring of a graph is an assignment of $r$ colors to the edges of the graph. An exactly $r$-edge-coloring of a graph is an $r$-edge-coloring of the graph that uses all $r$ colors. A matching of an edge-colored graph is called rainbow matching, if no two edges have the same color in the matching. In this paper, we prove that an exactly $r$-edge-colored complete graph of order $n$ has a rainbow matching of size $k(\ge 2)$ if $r \ge max\{{2k-3\choose 2}+2, {k-2\choose 2}+(k-2)(n-k+2)+2 \}$, $k \ge 2$, and $n \ge 2k+1$. The bound on $r$ is best possible.

Circular chromatic index of Cartesian products of graphs

AbstractThe circular chromatic index of a graph G, written $\chi_{c}'(G)$, is the minimum r permitting a function $f : E(G)\rightarrow [0,r)$ such that $1 \le | f(e)-f(e')|\le r - 1$ whenever e and $e'$ are incident. Let $G = H$ □ $C_{2m +1}$, where □ denotes Cartesian product and H is an $(s-2)$‐regular graph of odd order, with $s \equiv 0 \, {\rm mod}\, 4$ (thus, G is s‐regular). We prove that $\chi_{c}'(G) \ge s +\lfloor \lambda(1 - 1/s)\rfloor^{-1}$, where $\lambda$ is the minimum, over all bases of the cycle space of H, of the maximum length of a cycle in the basis. When $H = C_{2k +1}$ and m is large, the lower bound is sharp. In particular, if $m \ge 3k + 1$, then $\chi_{c}'(C_{2k +1}$ □ $C_{2m + 1})=4 + \lceil {3k/2} \rceil^{-1}$, independent of m. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 7–18, 2008