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 \(C_m\).     

A group of continuous self-maps on a topological groupoid

The group of bisections of groupoids plays an important role in the study of Lie groupoids. In this paper another construction is introduced. Indeed, for a topological groupoid G, the set of all continuous self-maps f on G such that (x, f(x)) is a composable pair for every \(x\in G\), is denoted by \(S_G\). We show that \(S_G\) by a natural binary operation is a monoid. \(S_G(\alpha )\), the group of units in \(S_G\) precisely consists of those \(f\in S_G\) such that the map \(x\mapsto xf(x)\) is a bijection on G. Similar to the group of bisections, \(S_G(\alpha )\) acts on G from the right and on the space of continuous self-maps on G from the left. It is proved that \(S_G(\alpha )\) with the compact- open topology inherited from C(G, G) is a left topological group. For a compact Hausdorff groupoid G it is proved that the group of bisections of \(G^2\) is isomorphic to the group \(S_G(\alpha )\) and the group of transitive bisections of G, \(Bis_T(G)\), is embedded in \(S_G(\alpha )\), where \(G^2\) is the groupoid of all composable pairs.     

Partial permutation decoding for binary linear and $$Z_4$$-linear Hadamard codes

In this paper, s-\({\text {PD}}\)-sets of minimum size \(s+1\) for partial permutation decoding for the binary linear Hadamard code \(H_m\) of length \(2^m\), for all \(m\ge 4\) and \(2 \le s \le \lfloor {\frac{2^m}{1+m}}\rfloor -1\), are constructed. Moreover, recursive constructions to obtain s-\({\text {PD}}\)-sets of size \(l\ge s+1\) for \(H_{m+1}\) of length \(2^{m+1}\), from an s-\({\text {PD}}\)-set of the same size for \(H_m\), are also described. These results are generalized to find s-\({\text {PD}}\)-sets for the \({\mathbb {Z}}_4\)-linear Hadamard codes \(H_{\gamma , \delta }\) of length \(2^m\), \(m=\gamma +2\delta -1\), which are binary Hadamard codes (not necessarily linear) obtained as the Gray map image of quaternary linear codes of type \(2^\gamma 4^\delta \). Specifically, s-PD-sets of minimum size \(s+1\) for \(H_{\gamma , \delta }\), for all \(\delta \ge 3\) and \(2\le s \le \lfloor {\frac{2^{2\delta -2}}{\delta }}\rfloor -1\), are constructed and recursive constructions are described.     

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 \(k\ge 4,\, t\ge 1\).     

Packing Chromatic Number of Base-3 Sierpiński Graphs

The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that there exists a k-vertex coloring of G in which any two vertices receiving color i are at distance at least \(i+1\). Let \(S^n\) be the base-3 Sierpiński graph of dimension n. It is proved that \(\chi _{\rho }(S^1) = 3\), \(\chi _{\rho }(S^2) = 5\), \(\chi _{\rho }(S^3) = \chi _{\rho }(S^4) = 7\), and that \(8\le \chi _\rho (S^n) \le 9\) holds for any \(n\ge 5\).     

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.     

An <Emphasis Type="Italic">A</Emphasis>-invariant subspace for bipartite distance-regular graphs with exactly two irreducible <Emphasis Type="Italic">T</Emphasis>-modules with endpoint 2, both thin

Let \(\Gamma \) denote a bipartite distance-regular graph with vertex set X, diameter \(D \ge 4\), and valency \(k \ge 3\). Let \({{\mathbb {C}}}^X\) denote the vector space over \({{\mathbb {C}}}\) consisting of column vectors with entries in \({{\mathbb {C}}}\) and rows indexed by X. For \(z \in X\), let \({{\widehat{z}}}\) denote the vector in \({{\mathbb {C}}}^X\) with a 1 in the z-coordinate, and 0 in all other coordinates. Fix a vertex x of \(\Gamma \) and let \(T = T(x)\) denote the corresponding Terwilliger algebra. Assume that up to isomorphism there exist exactly two irreducible T-modules with endpoint 2, and they both are thin. Fix \(y \in X\) such that \(\partial (x,y)=2\), where \(\partial \) denotes path-length distance. For \(0 \le i,j \le D\) define \(w_{ij}=\sum {{\widehat{z}}}\), where the sum is over all \(z \in X\) such that \(\partial (x,z)=i\) and \(\partial (y,z)=j\). We define \(W=\mathrm{span}\{w_{ij} \mid 0 \le i,j \le D\}\). In this paper we consider the space \(MW=\mathrm{span}\{mw \mid m \in M, w \in W\}\), where M is the Bose–Mesner algebra of \(\Gamma \). We observe that MW is the minimal A-invariant subspace of \({{\mathbb {C}}}^X\) which contains W, where A is the adjacency matrix of \(\Gamma \). We show that \(4D-6 \le \mathrm{dim}(MW) \le 4D-2\). We display a basis for MW for each of these five cases, and we give the action of A on these bases.     

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.     

Relationships Between the 2-Metric Dimension and the 2-Adjacency Dimension in the Lexicographic Product of Graphs

Given a connected simple graph \(G=(V(G),E(G))\), a set \(S\subseteq V(G)\) is said to be a 2-metric generator for G if and only if for any pair of different vertices \(u,v\in V(G)\), there exist at least two vertices \(w_1,w_2\in S\) such that \(d_G(u,w_i)\ne d_G(v,w_i)\), for every \(i\in \{1,2\}\), where \(d_G(x,y)\) is the length of a shortest path between x and y. The minimum cardinality of a 2-metric generator is the 2-metric dimension of G, denoted by \(\dim _2(G)\). The metric \(d_{G,2}: V(G)\times V(G)\longmapsto {\mathbb {N}}\cup \{0\}\) is defined as \(d_{G,2}(x,y)=\min \{d_G(x,y),2\}\). Now, a set \(S\subseteq V(G)\) is a 2-adjacency generator for G, if for every two vertices \(x,y\in V(G)\) there exist at least two vertices \(w_1,w_2\in S\), such that \(d_{G,2}(x,w_i)\ne d_{G,2}(y,w_i)\) for every \(i\in \{1,2\}\). The minimum cardinality of a 2-adjacency generator is the 2-adjacency dimension of G, denoted by \({\mathrm {adim}}_2(G)\). In this article, we obtain closed formulae for the 2-metric dimension of the lexicographic product \(G\circ H\) of two graphs G and H. Specifically, we show that \(\dim _2(G\circ H)=n\cdot {\mathrm {adim}}_2(H)+f(G,H),\) where \(f(G,H)\ge 0\), and determine all the possible values of f(G, H).     

Distance Magic Labeling in Complete 4-partite Graphs

Let G be a complete k-partite simple undirected graph with parts of sizes \(p_1\le p_2\cdots \le p_k\). Let \(P_j=\sum _{i=1}^jp_i\) for \(j=1,\ldots ,k\). It is conjectured that G has distance magic labeling if and only if \(\sum _{i=1}^{P_j} (n-i+1)\ge j{{n+1}\atopwithdelims (){2}}/k\) for all \(j=1,\ldots ,k\). The conjecture is proved for \(k=4\), extending earlier results for \(k=2,3\).     

Monochromatic Solutions for Multi-Term Unknowns

Given integers \(k\ge 2\), \(n \ge 2\), \(m \ge 2\) and \( a_1,a_2,\ldots ,a_m \in {\mathbb {Z}}{\backslash }{\{0\}}\), and let \(f(z)= \sum _{j=0}^{n}c_jz^j\) be a polynomial of integer coefficients with \(c_n>0\) and \((\sum _{i=1}^ma_i)|f(z)\) for some integer z. For a k-coloring of \([N]=\{1,2,\ldots ,N\}\), we say that there is a monochromatic solution of the equation \(a_1x_1+a_2x_2+\cdots +a_mx_m=f(z)\) if there exist pairwise distinct \(x_1,x_2,\ldots ,x_m\in [N]\) all of the same color such that the equation holds for some \(z\in \mathbb {Z}\). Problems of this type are often referred to as Ramsey-type problems. In this paper, it is shown that if \(a_i>0\) for \(1\le i\le m\), then there exists an integer \(N_0=N(k,m,n)\) such that for \(N\ge N_0\), each k-coloring of [N] contains a monochromatic solution \(x_1,x_2,\ldots ,x_m\) of the equation \(a_1x_1+a_2x_2+ \cdots +a_mx_m= f(z)\). Moreover, if n is odd and there are \(a_i\) and \(a_j\) such that \(a_ia_j<0\) for some \(1 \le i\ne j\le m\), then the assertion holds similarly.     

The Terwilliger algebra of the incidence graph of the Hamming graph

Let \(\varGamma \) be a distance-semiregular graph on Y, and let \(D^Y\) be the diameter of \(\varGamma \) on Y. Let \(\varDelta \) be the halved graph of \(\varGamma \) on Y. Fix \(x \in Y\). Let T and \(T'\) be the Terwilliger algebras of \(\varGamma \) and \(\varDelta \) with respect to x, respectively. Assume, for an integer i with \(1 \le 2i \le D^Y\) and for \(y,z \in \varGamma _{2i}(x)\) with \(\partial _{\varGamma }(y,z)=2\), the numbers \(|\varGamma _{2i-1}(x) \cap \varGamma (y) \cap \varGamma (z)|\) and \(|\varGamma _{2i+1}(x) \cap \varGamma (y) \cap \varGamma (z)|\) depend only on i and do not depend on the choice of y, z. The first goal in this paper is to show the relations between T-modules of \(\varGamma \) and \(T'\)-modules of \(\varDelta \). Assume \(\varGamma \) is the incidence graph of the Hamming graph H(D, n) on the vertex set Y and the set \({\mathcal {C}}\) of all maximal cliques. Then, \(\varGamma \) satisfies above assumption and \(\varDelta \) is isomorphic to H(D, n). The second goal is to determine the irreducible T-modules of \(\varGamma \). For each irreducible T-module W, we give a basis for W the action of the adjacency matrix on this basis and we calculate the multiplicity of W.     

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 \(A_5(7,5)\le 87\).     

Heinz mean curvature estimates in warped product spaces $$M\times _{e^{\psi }}N$$

If a graph submanifold (x, f(x)) of a Riemannian warped product space \((M^m\times _{e^{\psi }}N^n,\tilde{g}=g+ e^{2\psi }h)\) is immersed with parallel mean curvature H, then we obtain a Heinz-type estimation of the mean curvature. Namely, on each compact domain D of M, \(m\Vert H\Vert \le \frac{A_{\psi }(\partial D)}{V_{\psi }(D)}\) holds, where \(A_{\psi }(\partial D)\) and \(V_{\psi }(D)\) are the \({\psi }\)-weighted area and volume, respectively. In particular, \(H=0\) if (M, g) has zero-weighted Cheeger constant, a concept recently introduced by Impera et al. (Height estimates for killing graphs. arXiv:1612.01257, 2016). This generalizes the known cases \(n=1\) or \(\psi =0\). We also conclude minimality using a closed calibration, assuming \((M,g_*)\) is complete where \(g_*=g+e^{2\psi }f^*h\), and for some constants \(\alpha \ge \delta \ge 0\), \(C_1>0\) and \(\beta \in [0,1)\), \(\Vert \nabla ^*\psi \Vert ^2_{g_*}\le \delta \), \(\mathrm {Ricci}_{\psi ,g_*}\ge \alpha \), and \({\mathrm{det}}_g(g_*)\le C_1 r^{2\beta }\) holds when \(r\rightarrow +\infty \), where r(x) is the distance function on \((M,g_*)\) from some fixed point. Both results rely on expressing the squared norm of the mean curvature as a weighted divergence of a suitable vector field.     

Generalizations of Marriage Theorem for Degree Factors

Let G be a bipartite graph with bipartition (A, B). We give new criteria for a bipartite graph to have an f -factor, a (g, f)-factor and other factors together with some applications of these criteria. These criteria can be considered as direct generalizations of Hall’s marriage theorem. Among some results, we prove that for a function \(h: A\cup B \rightarrow \{0,1,2, \ldots \}\), G has a factor F such that \(\deg _F(x)=h(x)\) for \(x\in A\) and \(\deg _H(y) \le h(y)\) for \(y\in B\) if and only if \(h(X) \le \sum _{x\in N_G(X)}\min \{h(x), e_G(x,X)\}\) for all \(X\subseteq A\).     

On the family of elliptic curves $$\varvec{y^2=x^3-m^2x+p^2}$$

In this paper, we study the torsion subgroup and rank of elliptic curves for the subfamilies of \(E_{m,p} : y^2=x^3-m^2x+p^2\), where m is a positive integer and p is a prime. We prove that for any prime p, the torsion subgroup of \(E_{m,p}(\mathbb {Q})\) is trivial for both the cases {\(m\ge 1\), \(m\not \equiv 0\pmod 3\)} and {\(m\ge 1\), \(m \equiv 0 \pmod 3\), with \(gcd(m,p)=1\)}. We also show that given any odd prime p and for any positive integer m with \(m\not \equiv 0\pmod 3\) and \(m\equiv 2\pmod {32}\), the lower bound for the rank of \(E_{m,p}(\mathbb {Q})\) is 2. Finally, we find curves of rank 9 in this family.     

A new theorem on the prime-counting function

For \(x>0\), let \(\pi (x)\) denote the number of primes not exceeding x. For integers a and \(m>0\), we determine when there is an integer \(n>1\) with \(\pi (n)=(n+a)/m\). In particular, we show that, for any integers \(m>2\) and \(a\leqslant \lceil e^{m-1}/(m-1)\rceil \), there is an integer \(n>1\) with \(\pi (n)=(n+a)/m\). Consequently, for any integer \(m>4\), there is a positive integer n with \(\pi (mn)=m+n\). We also pose several conjectures for further research; for example, we conjecture that, for each \(m=1,2,3,\ldots \), there is a positive integer n such that \(m+n\) divides \(p_m+p_n\), where \(p_k\) denotes the k-th prime.     

Relative Difference Sets and Hadamard Matrices from Perfect Quaternionic Arrays

Let \(G=\mathbf{C}_{n_1}\times \cdots \times \mathbf{C}_{n_m}\) be an abelian group of order \(n=n_1\dots n_m\), where each \(\mathbf{C}_{n_t}\) is cyclic of order \(n_t\). We present a correspondence between the (4n, 2, 4n, 2n)-relative difference sets in \(G\times Q_8\) relative to the centre \(Z(Q_8)\) and the perfect arrays of size \(n_1\times \dots \times n_m\) over the quaternionic alphabet \(Q_8\cup qQ_8\), where \(q=(1+i+j+k)/2\). In view of this connection, for \(m=2\) we introduce new families of relative difference sets in \(G\times Q_8\), as well as new families of Williamson and Ito Hadamard matrices with G-invariant components.     

Strong extensions for q-summing o perators acting in p-convex Banach function s paces for $$1 \le p \le q$$1≤ p≤ q

Let \(1\le p\le q<\infty \) and let X be a p-convex Banach function space over a \(\sigma \)-finite measure \(\mu \). We combine the structure of the spaces \(L^p(\mu )\) and \(L^q(\xi )\) for constructing the new space \(S_{X_p}^{\,q}(\xi )\), where \(\xi \) is a probability Radon measure on a certain compact set associated to X. We show some of its properties, and the relevant fact that every q-summing operator T defined on X can be continuously (strongly) extended to \(S_{X_p}^{\,q}(\xi )\). Our arguments lead to a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provided the known (strong) factorizations for q-summing operators through \(L^q\)-spaces when \(1 \le q \le p\). Thus, our result completes the picture, showing what happens in the complementary case \(1\le p\le q\).     

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.