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For positive integers $ngeq kgeq t$, a collection $ mathcal{B} $ of $k$-subsets of an $n$-set $ X $ is called a $t$-packing if every $t$-subset of $ X $ appears in at most one set in $mathcal{B}$. In this paper, we give some upper and lower bounds for the maximum size of $3$-packings when $n$ is sufficiently larger than $k$. In one case, the upper and lower bounds are equal, in some cases, they differ by at most an additive constant depending only on $k$ and in one case they differ by a linear bound in $ n $.
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By rectangle packing we mean putting a set of rectangles into an enclosing rectangle, without any overlapping. We begin with perfect rectangle packing problems, then prove two continuity properties for parallel rectangle packing problems, and discuss how they might be used to obtain negative results for perfect rectangle packing problems.
Let $G$ be a graph of order $n(G)$ and vertex set $V(G)$. Given a set $Ssubseteq V(G)$, we define the external neighbourhood of $S$ as the set $N_e(S)$ of all vertices in $V(G)setminus S$ having at least one neighbour in $S$. The differential of $S$ is defined to be $partial(S)=|N_e(S)|-|S|$. In this paper, we introduce the study of the $2$-packing differential of a graph, which we define as $partial_{2p}(G)=max{partial(S): Ssubseteq V(G) text{ is a }2text{-packing}}.$ We show that the $2$-packing differential is closely related to several graph parameters, including the packing number, the independent domination number, the total domination number, the perfect differential, and the unique response Roman domination number. In particular, we show that the theory of $2$-packing differentials is an appropriate framework to investigate the unique response Roman domination number of a graph without the use of functions. Among other results, we obtain a Gallai-type theorem, which states that $partial_{2p}(G)+mu_{_R}(G)=n(G)$, where $mu_{_R}(G)$ denotes the unique response Roman domination number of $G$. As a consequence of the study, we derive several combinatorial results on $mu_{_R}(G)$, and we show that the problem of finding this parameter is NP-hard. In addition, the particular case of lexicographic product graphs is discussed.
We prove that there is $c>0$ such that for all sufficiently large $n$, if $T_1,dots,T_n$ are any trees such that $T_i$ has $i$ vertices and maximum degree at most $cn/log n$, then ${T_1,dots,T_n}$ packs into $K_n$. Our main result actually allows to replace the host graph $K_n$ by an arbitrary quasirandom graph, and to generalize from trees to graphs of bounded degeneracy that are rich in bare paths, contain some odd degree vertices, and only satisfy much less stringent restrictions on their number of vertices.
A linear system is a pair $(P,mathcal{L})$ where $mathcal{L}$ is a family of subsets on a ground finite set $P$, such that $|lcap l^prime|leq 1$, for every $l,l^prime in mathcal{L}$. The elements of $P$ and $mathcal{L}$ are called points and lines, respectively, and the linear system is called intersecting if any pair of lines intersect in exactly one point. A subset $T$ of points of $P$ is a transversal of $(P,mathcal{L})$ if $T$ intersects any line, and the transversal number, $tau(P,mathcal{L})$, is the minimum order of a transversal. On the other hand, a 2-packing set of a linear system $(P,mathcal{L})$ is a set $R$ of lines, such that any three of them have a common point, then the 2-packing number of $(P,mathcal{L})$, $ u_2(P,mathcal{L})$, is the size of a maximum 2-packing set. It is known that the transversal number $tau(P,mathcal{L})$ is bounded above by a quadratic function of $ u_2(P,mathcal{L})$. An open problem is to haracterize the families of linear systems which satisfies $tau(P,mathcal{L})leq lambda u_2(P,mathcal{L})$, for some $lambdageq1$. In this paper, we give an infinite family of linear systems $(P,mathcal{L})$ which satisfies $tau(P,mathcal{L})= u_2(P,mathcal{L})$ with smallest possible cardinality of $mathcal{L}$, as well as some properties of $r$-uniform intersecting linear systems $(P,mathcal{L})$, such that $tau(P,mathcal{L})= u_2(P,mathcal{L})=r$. Moreover, we state a characterization of $4$-uniform intersecting linear systems $(P,mathcal{L})$ with $tau(P,mathcal{L})= u_2(P,mathcal{L})=4$.
Let $p_{k,3}(n)$ enumerate the number of 2-color partition triples of $n$ where one of the colors appears only in parts that are multiples of $k$. In this paper, we prove several infinite families of congruences modulo powers of 3 for $p_{k,3}(n)$ with $k=1, 3$, and $9$. For example, for all integers $ngeq0$ and $alphageq1$, we prove that begin{align*} p_{3,3}left(3^{alpha}n+dfrac{3^{alpha}+1}{2}right) &equiv0pmod{3^{alpha+1}} end{align*} and begin{align*} p_{3,3}left(3^{alpha+1}n+dfrac{5times3^{alpha}+1}{2}right) &equiv0pmod{3^{alpha+4}}. end{align*}
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