Maximal sets of Hamilton cycles in Knr;λ1,λ2

Let $K\left(n^r;{\lambda }_1,{\lambda }_2\right)$ be the $r$-partite multigraph in which each part has size $n$, where two vertices in the same part or different parts are joined by exactly ${\lambda }_1$ edges or ${\lambda }_2$ edges, respectively. It is proved that there exists a maximal set of $t$ edge-disjoint Hamilton cycles in $K\left(n^r;{\lambda }_1,{\lambda }_2\right)$ for ${\lambda }_{2}n\lfloor \frac{r+3}{4}\rfloor \le t\le min\left\{\lfloor \frac{{\lambda }_2{n}^2(r-1)}{2}\rfloor ,\lfloor \frac{{\lambda }_1(n-1)+{\lambda }_{2}n(r-1)}{2}\rfloor \right\}$, the upper bound being best possible. The results proved make use of the method of amalgamations.     

On the asymptotic growth of bipartite graceful permutations

We give exact growth rates for the number of bipartite graceful permutations of the symbols $\{0,1,\dots ,n?1\}$ that start with $a$ for $a\le 14$ (equivalently, $\alpha$-labelings of paths with $n$ vertices that have $a$ as a pendant label). In particular, when $a=14$ the growth is asymptotically like ${\lambda }^n$ for $\lambda \approx 3.461$. The number of graceful permutations of length $n$ grows at least this fast, improving on the best existing asymptotic lower bound of $2.37^n$. Combined with existing theory, this improves the known lower bounds on the number of Hamiltonian decompositions of the complete graph $K_{2n+1}$ and on the number of cyclic oriented triangular embeddings of $K_{12s+3}$ and $K_{12s+7}$. We also give the first exponential lower bound on the number of R-sequencings of ${\mathbb{Z}}_{2n+1}$.     

On 2-walk-regular graphs with a large intersection number c2

For a connected amply regular graph with parameters $(v,k,\lambda ,\mu )$ satisfying $\lambda \le \mu$, it is known that its diameter is bounded by $k$. This was generalized by Terwilliger to $(s,c,a,k)$-graphs satisfying $c\ge max\{a,2\}$. It follows from Terwilliger that a connected amply regular graph with parameters $(v,k,\lambda ,\mu )$ satisfying $\mu >\frac{k}{3}\ge 1$ and $\mu \ge \lambda$ has diameter at most 7.In this paper we will classify the 2-walk-regular graphs with valency $k\ge 3$ and diameter at least 4 such that its intersection number $c_2$ satisfies $c_2>\frac{k}{3}$. This result generalizes a result of Koolen and Park for distance-regular graphs. And we show that if such a 2-walk-regular graph is not distance-regular, then it is the incidence graph of a group divisible design with the dual property with parameters $(2,m;k;0,{\lambda }_2)$.     

Symmetric designs and four dimensional projective special unitary groups

In this article, we study symmetric $(v,k,\lambda )$ designs admitting a flag-transitive and point-primitive automorphism group $G$ whose socle is ${\mathrm{PSU}}_4\left(q\right)$. We prove that there exist eight non-isomorphic such designs for which $\lambda \in \{3,6,18\}$ and $G$ is either ${\mathrm{PSU}}_4\left(2\right)$, or ${\mathrm{PSU}}_4\left(2\right):2$.     

A C5-decomposition of the λ-fold line graph of the complete graph

In 1996, Cox and Rodger [Cycle systems of the line graph of the complete graph, J. Graph Theory 21 (1996) 173–182] raised the following question: For what values of $m$ and $n$ does there exist an $m$-cycle decomposition of $L\left(K_n\right)?$ In this paper, the above question is answered for $m=5.$ In fact, it is shown that $L\left(K_n\right)\left(\lambda \right),$ the $\lambda$-fold line graph of the complete graph $K_n,$ has a $C_5$-decomposition if and only if $5\mid \lambda \left(\genfrac{}{}{0ex}{}{n}{2}\right)(n?2)$ and $n\ge 4.$     

Equitable 2-partitions of the Hamming graphs with the second eigenvalue

The eigenvalues of the Hamming graph $H(n,q)$ are known to be ${\lambda }_i(n,q)=(q-1)n-qi$, $0\le i\le n$. The characterization of equitable 2-partitions of the Hamming graphs $H(n,q)$ with eigenvalue ${\lambda }_1(n,q)$ was obtained by Meyerowitz (2003). We study the equitable 2-partitions of $H(n,q)$ with eigenvalue ${\lambda }_2(n,q)$. We show that these partitions are reduced to equitable 2-partitions of $H(3,q)$ with eigenvalue ${\lambda }_2(3,q)$ with the exception of two constructions.     

Enclosings of decompositions of complete multigraphs in 2-edge-connected r-factorizations

A decomposition of a multigraph $G$ is a partition of its edges into subgraphs $G\left(1\right),\dots ,G\left(k\right)$. It is called an $r$-factorization if every $G\left(i\right)$ is $r$-regular and spanning. If $G$ is a subgraph of $H$, a decomposition of $G$ is said to be enclosed in a decomposition of $H$ if, for every $1\le i\le k$, $G\left(i\right)$ is a subgraph of $H\left(i\right)$.Feghali and Johnson gave necessary and sufficient conditions for a given decomposition of $\lambda K_n$ to be enclosed in some 2-edge-connected $r$-factorization of $\mu K_m$ for some range of values for the parameters $n$, $m$, $\lambda$, $\mu$, $r$: $r=2$, $\mu >\lambda$ and either $m\ge 2n?1$, or $m=2n?2$ and $\mu =2$ and $\lambda =1$, or $n=3$ and $m=4$. We generalize their result to every $r\ge 2$ and $m\ge 2n?2$. We also give some sufficient conditions for enclosing a given decomposition of $\lambda K_n$ in some 2-edge-connected $r$-factorization of $\mu K_m$ for every $r\ge 3$ and $m>(2?C)n$, where $C$ is a constant that depends only on $r$, $\lambda$ and $\mu$.     

Spanning trees with at most 4 leaves in K1,5-free graphs

In 2009, Kyaw proved that every $n$-vertex connected $K_{1,4}$-free graph $G$ with ${\sigma }_4\left(G\right)\ge n?1$ contains a spanning tree with at most 3 leaves. In this paper, we prove an analogue of Kyaw’s result for connected $K_{1,5}$-free graphs. We show that every $n$-vertex connected $K_{1,5}$-free graph $G$ with ${\sigma }_5\left(G\right)\ge n?1$ contains a spanning tree with at most 4 leaves. Moreover, the degree sum condition “${\sigma }_5\left(G\right)\ge n?1$” is best possible.     

New upper bound for multicolor Ramsey number of odd cycles

Let $r_k\left(C_{2m+1}\right)$ be the $k$-color Ramsey number of an odd cycle $C_{2m+1}$ of length $2m+1$. It is shown that for each fixed $m\ge 2$, $r_k\left(C_{2m+1}\right)<c^k\sqrt{k!}$for all sufficiently large $k$, where $c=c\left(m\right)>0$ is a constant. This improves an old result by Bondy and Erd?s (1973).     

Erdős–Gallai stability theorem for linear forests

The Erd?s–Gallai Theorem states that every graph of average degree more than $l?2$ contains a path of order $l$ for $l\ge 2$. In this paper, we obtain a stability version of the Erd?s–Gallai Theorem in terms of minimum degree. Let $G$ be a connected graph of order $n$ and $F=\left({?}_{i=1}^k{P}_{2a_i}\right)?\left({?}_{i=1}^l{P}_{2b_i+1}\right)$ be $k+l$ disjoint paths of order $2a_1,\dots ,2a_k,2b_1+1,\dots ,2b_l+1,$ respectively, where $k\ge 0$, $0\le l\le 2$, and $k+l\ge 2$. If the minimum degree $\delta \left(G\right)\ge {\sum }_{i=1}^k{a}_i+{\sum }_{i=1}^l{b}_i?1$, then $F?G$ except several classes of graphs for sufficiently large $n$, which extends and strengths the results of Ali and Staton for an even path and Yuan and Nikiforov for an odd path.     

Blow-up phenomena for the Liouville equation with a singular source of integer multiplicity

We are concerned with the existence of blowing-up solutions to the following boundary value problem

$?\mathrm{\Delta }u=\lambda a(x)e^u?4\pi N{\delta }_0\text{ in }\mathrm{\Omega },u=0\text{ on }?\mathrm{\Omega },$

where Ω is a smooth and bounded domain in ${\mathbb{R}}^2$ such that $0\in \mathrm{\Omega }$, $a(x)$ is a positive smooth function, N is a positive integer and $\lambda >0$ is a small parameter. Here ${\delta }_0$ defines the Dirac measure with pole at 0. We find conditions on the function a and on the domain Ω under which there exists a solution $u_{\lambda }$ blowing up at 0 and satisfying $\lambda {\int }_{\mathrm{\Omega }}a(x)e^{u_{\lambda }}\to 8\pi (N+1)$ as $\lambda \to 0^{+}$.     

The covering radius of the Reed–Muller code RM(2,7) is 40

It was proved by J. Schatz that the covering radius of the second order Reed–Muller code $RM(2,6)$ is 18 (Schatz (1981)). However, the covering radius of $RM(2,7)$ has been an open problem for many years. In this paper, we prove that the covering radius of $RM(2,7)$ is 40, which is the same as the covering radius of $RM(2,7)$ in $RM(3,7)$. As a corollary, we also find new upper bounds for the covering radius of $RM(2,n)$, $n=8,9,10$.     

The Nehari manifold of biharmonic equations with p-Laplacian and singular potential

In this paper, we investigate a class of biharmonic equations with $p$-Laplacian and singular potential as follows: $\left\{\begin{array}{cc}{\Delta }^{2}u+V_{\lambda }\left(x\right)u+\text{d}iv\left(\rho \left(x\right)\right|\nabla u{|}^{p-2}\nabla u)=0\hfill & \text{in}{\mathbb{R}}^N,\hfill \\ u\in H^2\left({\mathbb{R}}^N\right),\hfill & \hfill \end{array}\right.$where $N\ge 3,$ $1<p<\frac{2N}{N-2}$ except $p=2$ and $V_{\lambda }\left(x\right)=\lambda a\left(x\right)-b\left(x\right)$ with $\lambda >0$. Under some suitable assumptions on $a,b$ and $\rho$, by using the Nehari manifold, we obtain the existence of nontrivial solutions for $\lambda$ large enough which improves the existing result in the literature.     

Small time convergence of subordinators with regularly or slowly varying canonical measure

We consider subordinators $X_{\alpha }={\left(X_{\alpha }\left(t\right)\right)}_{t\ge 0}$ in the domain of attraction at 0 of a stable subordinator ${\left(S_{\alpha }\left(t\right)\right)}_{t\ge 0}$ (where $\alpha \in (0,1)$); thus, with the property that ${\overline{\Pi }}_{\alpha }$, the tail function of the canonical measure of $X_{\alpha }$, is regularly varying of index $?\alpha \in (?1,0)$ as $x\downarrow 0$. We also analyse the boundary case, $\alpha =0$, when ${\overline{\Pi }}_{\alpha }$ is slowly varying at 0. When $\alpha \in (0,1)$, we show that ${\left(t{\overline{\Pi }}_{\alpha }\left(X_{\alpha }\left(t\right)\right)\right)}^{?1}$ converges in distribution, as $t\downarrow 0$, to the random variable ${\left(S_{\alpha }\left(1\right)\right)}^{\alpha }$. This latter random variable, as a function of $\alpha$, converges in distribution as $\alpha \downarrow 0$ to the inverse of an exponential random variable. We prove these convergences, also generalised to functional versions (convergence in $\mathbb{D}[0,1]$), and to trimmed versions, whereby a fixed number of its largest jumps up to a specified time are subtracted from the process. The $\alpha =0$ case produces convergence to an extremal process constructed from ordered jumps of a Cauchy subordinator. Our results generalise random walk and stable process results of Darling, Cressie, Kasahara, Kotani and Watanabe.