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

Corrádi 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 Corrádi–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 Corrádi–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.     

On a problem of Erdős and Moser

A set A of vertices in an r-uniform hypergraph \(\mathcal H\) is covered in \(\mathcal H\) if there is some vertex \(u\not \in A\) such that every edge of the form \(\{u\}\cup B\), \(B\in A^{(r-1)}\) is in \(\mathcal H\). Erd?s and Moser (J Aust Math Soc 11:42–47, 1970) determined the minimum number of edges in a graph on n vertices such that every k-set is covered. We extend this result to r-uniform hypergraphs on sufficiently many vertices, and determine the extremal hypergraphs. We also address the problem for directed graphs.     

Cliques in $$C_4$$-free graphs of large minimum degree

A graph G is called \(C_4\)-free if it does not contain the cycle \(C_4\) as an induced subgraph. Hubenko, Solymosi and the first author proved (answering a question of Erd?s) a peculiar property of \(C_4\)-free graphs: \(C_4\)-free graphs with n vertices and average degree at least cn contain a complete subgraph (clique) of size at least \(c'n\) (with \(c'= 0.1c^2\)). We prove here better bounds \(\big ({c^2n\over 2+c}\) in general and \((c-1/3)n\) when \( c \le 0.733\big )\) from the stronger assumption that the \(C_4\)-free graphs have minimum degree at least cn. Our main result is a theorem for regular graphs, conjectured in the paper mentioned above: 2k-regular \(C_4\)-free graphs on \(4k+1\) vertices contain a clique of size \(k+1\). This is the best possible as shown by the kth power of the cycle \(C_{4k+1}\).     

Anti-Ramsey Numbers of Graphs with Small Connected Components

The anti-Ramsey number, AR(n, G), for a graph G and an integer \(n\ge |V(G)|\), is defined to be the minimal integer r such that in any edge-colouring of \(K_n\) by at least r colours there is a multicoloured copy of G, namely, a copy of G that each of its edges has a distinct colour. In this paper we determine, for large enough \(n,\, AR(n,L\cup tP_2)\) and \(AR(n,L\cup kP_3)\) for any large enough t and k, and a graph L satisfying some conditions. Consequently, we determine AR(n, G), for large enough n, where G is \(P_3\cup tP_2\) for any \(t\ge 3,\, P_4\cup tP_2\) and \(C_3\cup tP_2\) for any \(t\ge 2,\, kP_3\) for any \(k\ge 3,\, tP_2\cup kP_3\) for any \(t\ge 1,\, k\ge 2\), and \(P_{t+1}\cup kP_3\) for any \(t\ge 3,\, k\ge 1\). Furthermore, we obtain upper and lower bounds for AR(n, G), for large enough n, where G is \(P_{k+1}\cup tP_2\) and \(C_k\cup tP_2\) for any \(k\ge 4,\, t\ge 1\).     

Beatty sequences and prime numbers with restrictions on strongly <Emphasis Type="Italic">q</Emphasis>-additive functions

We estimate exponential sums over a non-homogenous Beatty sequence with restriction on strongly q-additive functions. We then apply our result in a few special cases to obtain an asymptotic formula for the number of primes \(p=\lfloor \alpha n +\beta \rfloor \) and \(f(p)\equiv a (\mathrm{mod\,}b)\), with \(n\ge N \), where \(\alpha \), \(\beta \) are real numbers and f is a strongly q-additive function (for example, the sum of digits function in base q is a strongly q-additive function). We also prove that for any fixed integer \(k\ge 3 \), all sufficiently large \(N\equiv k (\mathrm{mod\,}2) \) could be represented as a sum of k prime numbers from a Beatty sequence with restriction on strongly q-additive functions.     

Quantum MDS codes with relatively large minimum distance from Hermitian self-orthogonal codes

It has become common knowledge that constructing q-ary quantum MDS codes with minimum distance bigger than \(q/2+1\) is significantly more difficult than constructing those with minimum distance less than or equal to \(q/2+1\). Despite of various constructions of q-ary quantum MDS codes, all known q-ary quantum MDS codes have minimum distance bounded by \(q/2+1\) except for some lengths. The purpose of the current paper is to provide some new q-ary quantum MDS codes with minimum distance bigger than \(q/2+1\). In this paper, we provide several classes of quantum MDS codes with minimum distance bigger than \(q/2+1\). For instance, some examples in these classes include q-ary \([n,n-2k, k+1]\)-quantum MDS codes for cases: (i) \(q\equiv -1\bmod {5}, n=(q^2+4)/5\) and \(1\le k\le (3q-2)/5\); (ii) \(q\equiv -1\bmod {7}, n=(q^2+6)/7\) and \(1\le k\le (4q-3)/7\); (iii) \(2|q, q\equiv -1\bmod {3}, n=2(q^2-1)/3\) and \(1\le k\le (2q-1)/3\); and (iv) \(2|q, q\equiv -1\bmod {5}, n=2(q^2-1)/5\) and \(1\le k\le (3q-2)/5\).     

Cyclic compositions and cycles of the hypercube

The partition graph of a positive integer n, \(P_n\), is the graph whose vertices are the cyclic compositions of n and two vertices are adjacent if one composition is obtained from the other one by replacing two cyclically consecutive parts by their sum. In this paper we introduce and investigate the notions of singular cyclic composition and singular edge of \(P_n\). We associate with every singular edge and every cycle of \(P_n\), whose vertices are aperiodic cyclic compositions of n, a cycle or a set of disjoint cycles of equal length of the hypercube \(Q_n\).     

Complexities of normal bases constructed from Gauss periods

Let q be a power of a prime p, and let \(r=nk+1\) be a prime such that \(r\not \mid q\), where n and k are positive integers. Under a simple condition on q, r and k, a Gauss period of type (n, k) is a normal element of \({\mathbb {F}}_{q}^{n}\) over \({\mathbb {F}}_q\); the complexity of the resulting normal basis of \({\mathbb {F}}_{q}^{n}\) over \({\mathbb {F}}_q\) is denoted by C(n, k; p). Recent works determined C(n, k; p) for \(k\le 7\) and all qualified n and q. In this paper, we show that for any given \(k>0\), C(n, k; p) is given by an explicit formula except for finitely many primes \(r=nk+1\) and the exceptional primes are easily determined. Moreover, we describe an algorithm that allows one to compute C(n, k; p) for the exceptional primes \(r=nk+1\). Our numerical results cover C(n, k; p) for \(k\le 20\) and all qualified n and q.     

2-Domination number of generalized Petersen graphs

Let \(G=(V,E)\) be a graph. A subset \(S\subseteq V\) is a k-dominating set of G if each vertex in \(V-S\) is adjacent to at least k vertices in S. The k-domination number of G is the cardinality of the smallest k-dominating set of G. In this paper, we shall prove that the 2-domination number of generalized Petersen graphs \(P(5k+1, 2)\) and \(P(5k+2, 2)\), for \(k>0\), is \(4k+2\) and \(4k+3\), respectively. This proves two conjectures due to Cheng (Ph.D. thesis, National Chiao Tung University, 2013). Moreover, we determine the exact 2-domination number of generalized Petersen graphs P(2k, k) and \(P(5k+4,3)\). Furthermore, we give a good lower and upper bounds on the 2-domination number of generalized Petersen graphs \(P(5k+1, 3), P(5k+2,3)\) and \(P(5k+3, 3).\)     

Coloring of the Square of Kneser Graph $$$$

The Kneser graph K(n, k) is the graph whose vertices are the k-element subsets of an n elements set, with two vertices adjacent if they are disjoint. The square \(G^2\) of a graph G is the graph defined on V(G) such that two vertices u and v are adjacent in \(G^2\) if the distance between u and v in G is at most 2. Determining the chromatic number of the square of the Kneser graph K(n, k) is an interesting graph coloring problem, and is also related with intersecting family problem. The square of K(2k, k) is a perfect matching and the square of K(n, k) is the complete graph when \(n \ge 3k-1\). Hence coloring of the square of \(K(2k +1, k)\) has been studied as the first nontrivial case. In this paper, we focus on the question of determining \(\chi (K^2(2k+r,k))\) for \(r \ge 2\). Recently, Kim and Park (Discrete Math 315:69–74, 2014) showed that \(\chi (K^2(2k+1,k)) \le 2k+2\) if \( 2k +1 = 2^t -1\) for some positive integer t. In this paper, we generalize the result by showing that for any integer r with \(1 \le r \le k -2\),

1. (a)
   \(\chi (K^2 (2k+r, k)) \le (2k+r)^r\),   if   \(2k + r = 2^t\) for some integer t, and
    
2. (b)
   \(\chi (K^2 (2k+r, k)) \le (2k+r+1)^r\),   if  \(2k + r = 2^t-1\) for some integer t.
    

On the other hand, it was shown in Kim and Park (Discrete Math 315:69–74, 2014) that \(\chi (K^2 (2k+r, k)) \le (r+2)(3k + \frac{3r+3}{2})^r\) for \(2 \le r \le k-2\). We improve these bounds by showing that for any integer r with \(2 \le r \le k -2\), we have \(\chi (K^2 (2k+r, k)) \le 2 \left( \frac{9}{4}k + \frac{9(r+3)}{8} \right) ^r\). Our approach is also related with injective coloring and coloring of Johnson graph.

     

Approximation algorithms for constructing spanning K-trees using stock pieces of bounded length

Given a weighted graph G on \(n + 1\) vertices, a spanning K-tree \(T_K\) of G is defined to be a spanning tree T of G together with K distinct edges of G that are not edges of T. The objective of the minimum-cost spanning K-tree problem is to choose a subset of edges to form a spanning K-tree with the minimum weight. In this paper, we consider the constructing spanning K-tree problem that is a generalization of the minimum-cost spanning K-tree problem. We are required to construct a spanning K-tree \(T_K\) whose \(n+K\) edges are assembled from some stock pieces of bounded length L. Let \(c_0\) be the sale price of each stock piece of length L and \(k(T_K)\) the number of minimum stock pieces to construct the \(n+K\) edges in \(T_K\). For each edge e in G, let c(e) be the construction cost of that edge e. Our new objective is to minimize the total cost of constructing a spanning K-tree \(T_K\), i.e., \(\min _{T_K}\{\sum _{e\in T_K} c(e)+ k(T_K)\cdot c_0\}\). The main results obtained in this paper are as follows. (1) A 2-approximation algorithm to solve the constructing spanning K-tree problem. (2) A \(\frac{3}{2}\)-approximation algorithm to solve the special case for constant construction cost of edges. (3) An APTAS for this special case.     

New Rates of Convergence for the Iterates of Some Positive Linear Operators

In this paper, we give quantitative results for the convergence of the iterates of some positive linear operators which preserve the functions \(e_0\) and \(e_1\), respectively, \(e_0\) and \(e_2\). We obtain estimates in terms of moduli of smoothness and we improve some previous results. We give examples for the q-Bernstein, the q-Bernstein-Durrmeyer, the q-Stancu, the q-Meyer-Konig–Zeller, the King type of the q-Bernstein and the King type of the q-genuine-Bernstein-Durrmeyer operators. We observe that some q-operators (\(0<q<1\)) provide better convergence for the iterates than the corresponding classical operators (\(q=1\)).     

Coloring Complete and Complete Bipartite Graphs from Random Lists

Assign to each vertex v of the complete graph \(K_n\) on n vertices a list L(v) of colors by choosing each list independently and uniformly at random from all f(n)-subsets of a color set \([n] = \{1,\dots , n\}\), where f(n) is some integer-valued function of n. Such a list assignment L is called a random (f(n), [n])-list assignment. In this paper, we determine the asymptotic probability (as \(n \rightarrow \infty \)) of the existence of a proper coloring \(\varphi \) of \(K_n\), such that \(\varphi (v) \in L(v)\) for every vertex v of \(K_n\). We show that this property exhibits a sharp threshold at \(f(n) = \log n\). Additionally, we consider the corresponding problem for the line graph of a complete bipartite graph \(K_{m,n}\) with parts of size m and n, respectively. We show that if \(m = o(\sqrt{n})\), \(f(n) \ge 2 \log n\), and L is a random (f(n), [n])-list assignment for the line graph of \(K_{m,n}\), then with probability tending to 1, as \(n \rightarrow \infty \), there is a proper coloring of the line graph of \(K_{m,n}\) with colors from the lists.     

Semidefinite bounds for nonbinary codes based on quadruples

For nonnegative integers q, n, d, let \(A_q(n,d)\) denote the maximum cardinality of a code of length n over an alphabet [q] with q letters and with minimum distance at least d. We consider the following upper bound on \(A_q(n,d)\). For any k, let \(\mathcal{C}_k\) be the collection of codes of cardinality at most k. Then \(A_q(n,d)\) is at most the maximum value of \(\sum _{v\in [q]^n}x(\{v\})\), where x is a function \(\mathcal{C}_4\rightarrow {\mathbb {R}}_+\) such that \(x(\emptyset )=1\) and \(x(C)=\!0\) if C has minimum distance less than d, and such that the \(\mathcal{C}_2\times \mathcal{C}_2\) matrix \((x(C\cup C'))_{C,C'\in \mathcal{C}_2}\) is positive semidefinite. By the symmetry of the problem, we can apply representation theory to reduce the problem to a semidefinite programming problem with order bounded by a polynomial in n. It yields the new upper bounds \(A_4(6,3)\le 176\), \(A_4(7,3)\le 596\), \(A_4(7,4)\le 155\), \(A_5(7,4)\le 489\), and \(A_5(7,5)\le 87\).     

On Star–Wheel Ramsey Numbers

For two given graphs \(G_1\) and \(G_2\), the Ramsey number \(R(G_1,G_2)\) is the least integer r such that for every graph G on r vertices, either G contains a \(G_1\) or \(\overline{G}\) contains a \(G_2\). In this note, we determined the Ramsey number \(R(K_{1,n},W_m)\) for even m with \(n+2\le m\le 2n-2\), where \(W_m\) is the wheel on \(m+1\) vertices, i.e., the graph obtained from a cycle \(C_m\) by adding a vertex v adjacent to all vertices of the \(C_m\).     

Approximate results for rainbow labelings

A simple graph \(G=(V,\,E)\) is said to be antimagic if there exists a bijection \(f{\text {:}}\,E\rightarrow [1,\,|E|]\) such that the sum of the values of f on edges incident to a vertex takes different values on distinct vertices. The graph G is distance antimagic if there exists a bijection \(f{\text {:}}\,V\rightarrow [1,\, |V|],\) such that \(\forall x,\,y\in V,\)

$$\begin{aligned} \sum _{x_i\in N(x)}f\left( x_i\right) \ne \sum _{x_j\in N(y)}f\left( x_j\right) . \end{aligned}$$

Using the polynomial method of Alon we prove that there are antimagic injections of any graph G with n vertices and m edges in the interval \([1,\,2n+m-4]\) and, for trees with k inner vertices, in the interval \([1,\,m+k].\) In particular, a tree all of whose inner vertices are adjacent to a leaf is antimagic. This gives a partial positive answer to a conjecture by Hartsfield and Ringel. We also show that there are distance antimagic injections of a graph G with order n and maximum degree \(\Delta \) in the interval \([1,\,n+t(n-t)],\) where \( t=\min \{\Delta ,\,\lfloor n/2\rfloor \},\) and, for trees with k leaves, in the interval \([1,\, 3n-4k].\) In particular, all trees with \(n=2k\) vertices and no pairs of leaves sharing their neighbour are distance antimagic, a partial solution to a conjecture of Arumugam.

     

Rainbow matchings in bipartite multigraphs

Suppose that k is a non-negative integer and a bipartite multigraph G is the union of

$$\begin{aligned} N=\left\lfloor \frac{k+2}{k+1}n\right\rfloor -(k+1) \end{aligned}$$

matchings \(M_1,\dots ,M_N\), each of size n. We show that G has a rainbow matching of size \(n-k\), i.e. a matching of size \(n-k\) with all edges coming from different \(M_i\)’s. Several choices of the parameter k relate to known results and conjectures.

     

Universal Cycle Packings and Coverings for <Emphasis Type="Italic">k</Emphasis>-Subsets of an <Emphasis Type="Italic">n</Emphasis>-Set

A cyclic sequence of elements of [n] is an (n, k)-Ucycle packing (respectively, (n, k)-Ucycle covering) if every k-subset of [n] appears in this sequence at most once (resp. at least once) as a subsequence of consecutive terms. Let \(p_{n,k}\) be the length of a longest (n, k)-Ucycle packing and \(c_{n,k}\) the length of a shortest (n, k)-Ucycle covering. We show that, for a fixed \(k,p_{n,k}={n\atopwithdelims ()k}-O(n^{\lfloor k/2\rfloor })\). Moreover, when k is not fixed, we prove that if \(k=k(n)\le n^{\alpha }\), where \(0<\alpha <1/3\), then \(p_{n,k}={n\atopwithdelims ()k}-o({n\atopwithdelims ()k}^\beta )\) and \(c_{n,k}={n\atopwithdelims ()k}+o({n\atopwithdelims ()k}^\beta )\), for some \(\beta <1\). Finally, we show that if \(k=o(n)\), then \(p_{n,k}={n\atopwithdelims ()k}(1-o(1))\).     

Subgroups of cyclic groups and values of the Riemann zeta function

Let k be a positive integer, x a large real number, and let \(C_n\) be the cyclic group of order n. For \(k\le n\le x\) we determine the mean average order of the subgroups of \(C_n\) generated by k distinct elements and we give asymptotic results of related averaging functions of the orders of subgroups of cyclic groups. The average order is expressed in terms of Jordan’s totient functions and Stirling numbers of the second kind. We have the following consequence. Let k and x be as above. For \(k\le n\le x\), the mean average proportion of \(C_n\) generated by k distinct elements approaches \(\zeta (k+2)/\zeta (k+1)\) as x grows, where \(\zeta (s)\) is the Riemann zeta function.