Let $k,p,q$ be three positive integers. A graph $G$ with order $n$ is said to be $k$-placeable if there are $k$ edge disjoint copies of $G$ in the complete graph on $n$ vertices. A $(p,\,q)$-graph is a graph of order $p$ with $q$ edges. Packing results have proved useful in the study of the complexity of graph properties. Bollobás et al. investigated the $k$-placeable of $(n,\,n-2)$-graphs and $(n,\,n-1)$-graphs with $k=2$ and $k=3$. Motivated by their results, this paper characterizes $(n,\,n-1)$-graphs with girth at least $9$ which are $4$-placeable. We also consider the $k$-placeable of $(n,\,n+1)$-graphs and 2-factors.
Let $P$ be a partially ordered set. We prove that if $n$ is sufficiently large, then there exists a packing $\mathcal{P}$ of copies of $P$ in the Boolean lattice $(2^{[n]},\subset)$ that covers almost every element of $2^{[n]}$: $\mathcal{P}$ might not cover the minimum and maximum of $2^{[n]}$, and at most $|P|-1$ additional points due to divisibility. In particular, if $|P|$ divides $2^{n}-2$, then the truncated Boolean lattice $2^{[n]}-\{\emptyset,[n]\}$ can be partitioned into copies of $P$. This confirms a conjecture of Lonc from 1991.
We prove that any quasirandom graph with $n$ vertices and $rn$ edges can be decomposed into $n$ copies of any fixed tree with $r$ edges. The case of decomposing a complete graph establishes a conjecture of Ringel from 1963.
Let $H$ be an induced subgraph of the torus $C_k^m$. We show that when $k \ge 3$ is even and $|V(H)|$ divides some power of $k$, then for sufficiently large $n$ the torus $C_k^n$ has a perfect vertex-packing with induced copies of $H$. On the other hand, disproving a conjecture of Gruslys, we show that when $k$ is odd and not a prime power, then there exists $H$ such that $|V(H)|$ divides some power of $k$, but there is no $n$ such that $C_k^n$ has a perfect vertex-packing with copies of $H$. We also disprove a conjecture of Gruslys, Leader and Tan by exhibiting a subgraph $H$ of the $k$-dimen…
Let $G$ and $H$ be graphs. We say that $P$ is an $H$-packing of $G$ if $P$ is a set of edge-disjoint copies of $H$ in $G$. An $H$-packing $P$ is maximal if there is no other $H$-packing of $G$ that properly contains $P$. Packings of maximum cardinality have been studied intensively, with several recent breakthrough results. Here, we consider minimum cardinality maximal packings. An $H$-packing $P$ is clumsy if it is maximal of minimum size. Let $cl(G,H)$ be the size of a clumsy $H$-packing of $G$. We provide nontrivial bounds for $cl(G,H)$, and in many cases asymptotically determine $cl(G,H)$ …
Erdős and Hanani proved that for every fixed integer $k \ge 2$, the complete graph $K_n$ can be almost completely packed with copies of $K_k$; that is, $K_n$ contains pairwise edge-disjoint copies of $K_k$ that cover all but an $o_n(1)$ fraction of its edges. Equivalently, elements of the set $\C(k)$ of all red-blue edge colorings of $K_k$ can be used to almost completely pack every red-blue edge coloring of $K_n$. The following strengthening of the aforementioned Erdős-Hanani result is considered. Suppose $\C' \subset \C(k)$. Is it true that we can use elements only from $\C'$ and almost comp…
A set $B$ of vertices in a graph $G$ is called a \emph{$k$-limited packing} if for each vertex $v$ of $G$, its closed neighbourhood has at most $k$ vertices in $B$. The \emph{$k$-limited packing number} of a graph $G$, denoted by $L_k(G)$, is the largest number of vertices in a $k$-limited packing in $G$. The concept of the $k$-limited packing of a graph was introduced by Gallant et al., which is a generalization of the well-known packing of a graph. In this paper, we present some tight bounds for the $k$-limited packing number of a graph in terms of its order, diameter, girth, and maximum deg…
Recently, Balogh, Kostochka and Liu in [Packing chromatic number of cubic graphs, Discrete Math.~341 (2018) 474--483] answered in negative the question that was posed in several earlier papers whether the packing chromatic number is bounded in the class of graphs with maximum degree $3$. In this note, we present an explicit infinite family of subcubic graphs with unbounded packing chromatic number.
The \textit{packing chromatic number} of a graph $G$, denoted by $% χ_ρ(G)$, is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in \{1,\ldots,k\}$, where each $V_i$ is an $i$-packing. In this paper, we present some general properties of packing chromatic numbers of \textit{finite super subdivisions} of graphs. We determine the packing chromatic numbers of the finite super subdivisions of complete graphs, cycles and \textit{neighborhood corona graphs} of a cycle and a path respectively of a complete graph and a path.
For a nondecreasing sequence of integers $S=(s_1, s_2, \ldots)$ an $S$-packing $k$-coloring of a graph $G$ is a mapping from $V(G)$ to $\{1, 2,\ldots,k\}$ such that vertices with color $i$ have pairwise distance greater than $s_i$. By setting $s_i = d + \lfloor \frac{i-1}{n} \rfloor$ we obtain a $(d,n)$-packing coloring of a graph $G$. The smallest integer $k$ for which there exists a $(d,n)$-packing coloring of $G$ is called the $(d,n)$-packing chromatic number of $G$. In the special case when $d$ and $n$ are both equal to one we speak of the packing chromatic number of $G$. We determine the …
For $1\leq s_1 \le s_2 \le \ldots \le s_k$ and a graph $G$, a packing $(s_1, s_2, \ldots, s_k)$-coloring of $G$ is a partition of $V(G)$ into sets $V_1, V_2, \ldots, V_k$ such that, for each $1\leq i \leq k$, the distance between any two distinct $x,y\in V_i$ is at least $s_i + 1$. The packing chromatic number, $χ_p(G)$, of a graph $G$ is the smallest $k$ such that $G$ has a packing $(1,2, \ldots, k)$-coloring. It is known that there are trees of maximum degree 4 and subcubic graphs $G$ with arbitrarily large $χ_p(G)$. Recently, there was a series of papers on packing $(s_1, s_2, \ldots, s_k)$…
Given two graphs $H$ and $F$, the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices is denoted by $ex(n,H,F)$. We investigate the function $ex(n,H,kF)$, where $kF$ denotes $k$ vertex disjoint copies of a fixed graph $F$. Our results include cases when $F$ is a complete graph, cycle or a complete bipartite graph.
We resolve a conjecture of Cox and Martin by determining asymptotically for every $k\ge 2$ the maximum number of copies of $C_{2k}$ in an $n$-vertex planar graph.
A set $S$ of vertices in a graph is an open packing if (open) neighborhoods of any two distinct vertices in $S$ are disjoint. In this paper, we consider the graphs that have a unique maximum open packing. We characterize the trees with this property by using four local operations such that any nontrivial tree with a unique maximum open packing can be obtained by a sequence of these operations starting from $P_2$. We also prove that the decision version of the open packing number is NP-complete even when restricted to graphs of girth at least $6$. Finally, we show that the recognition of the gr…
Given a graph $G$ and a nondecreasing sequence $S=(s_1,\ldots,s_k)$ of positive integers, the mapping $c:V(G)\longrightarrow \{1,\ldots,k\}$ is called an $S$-packing coloring of $G$ if for any two distinct vertices $x$ and $y$ in $c^{-1}(i)$, the distance between $x$ and $y$ is greater than $s_i$. The smallest integer $k$ such that there exists a $(1,2,\ldots,k)$-packing coloring of a graph $G$ is called the packing chromatic number of $G$, denoted $χ_ρ(G)$. The question of boundedness of the packing chromatic number in the class of subcubic (planar) graphs was investigated in several earlier …
The triangle packing number $ν(G)$ of a graph $G$ is the maximum size of a set of edge-disjoint triangles in $G$. Tuza conjectured that in any graph $G$ there exists a set of at most $2ν(G)$ edges intersecting every triangle in $G$. We show that Tuza's conjecture holds in the random graph $G=G(n,m)$, when $m \le 0.2403n^{3/2}$ or $m\ge 2.1243n^{3/2}$. This is done by analyzing a greedy algorithm for finding large triangle packings in random graphs.
We study the maximum number $ex(n,e,H)$ of copies of a graph $H$ in graphs with given number of vertices and edges. We show that for any fixed graph $H$, $ex(n,e,H)$ is asymptotically realized by the quasi-clique provided that the edge density is sufficiently large. We also investigate a variant of this problem, when the host graph is bipartite.
Given a graph $G$, a function $c:V(G)\longrightarrow \{1,\ldots,k\}$ with the property that $c(u)=c(v)=i$ implies that the distance between $u$ and $v$ is greater than $i$, is called a $k$-packing coloring of $G$. The smallest integer $k$ for which there exists a $k$-packing coloring of $G$ is called the packing chromatic number of $G$, and is denoted by $χ_ρ$. Packing chromatic vertex-critical graphs are the graphs $G$ for which $χ_ρ(G-x) < χ_ρ(G)$ holds for every vertex $x$ of $G$. A graph $G$ is called a packing chromatic critical graph if for every proper subgraph $H$ of $G$, $χ_ρ(H) < χ_ρ…
A dominating set of a graph $G$ is a set $D\subseteq V(G)$ such that \-every vertex of $G$ is either in $D$ or is adjacent to a vertex in $D$. The domination number of $G$, $γ(G)$, is the minimum order of a dominating set. A subset $R$ of edges of a graph $G$ is a 2-degree-packing, if any three edges from $R$ do not have the same incident vertex. The 2-degree-packing number of $G$, $ν_2(G)$, is the maximum order of a 2-degree-packing of $G$. In this paper, we prove that any simple graph $G$ satisfies $γ(G)\leqν_2(G)-1$. Furthermore, we give a characterization of simple connected graphs $G$ sat…
For graphs $H$ and $F$, let $\operatorname{ex}(n, H, F)$ be the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices. The study of this function, which generalises the well-studied Turán numbers of graphs, was initiated recently by Alon and Shikhelman. We show that if $F$ is a tree then $\operatorname{ex}(n, H, F) = Θ(n^r)$ for some integer $r = r(H, F)$, thus answering one of their questions.
We consider a natural generalisation of Turán's forbidden subgraph problem and the Ruzsa-Szemerédi problem by studying the maximum number $ex_F(n,G)$ of edge-disjoint copies of a fixed graph $F$ can be placed on an $n$-vertex ground set without forming a subgraph $G$ whose edges are from different $F$-copies. We determine the pairs $\{F, G\}$ for which the order of magnitude of $ex_F(n,G)$ is quadratic and prove several asymptotic results using various tools from the regularity lemma and supersaturation to graph packing results.
The recently introduced $\{k\}$-packing function problem is considered in this paper. Special relation between a case when $k=1$, $k\ge 2$ and linear programming relaxation is introduced with sufficient conditions for optimality. For arbitrary simple connected graph $G$ there is construction procedure for finding values of $k$ for which $L_{\{k\}}(G)$ can be determined in the polynomial time. Additionally, relationship between $\{1\}$-packing function and independent set number is established. Optimal values for some special classes of graphs and general upper and lower bounds are introduced.
We consider a planar graph $G$ in which the edges have nonnegative integer lengths such that the length of every cycle of $G$ is even, and three faces are distinguished, called holes in $G$. It is known that there exists a packing of cuts and (2,3)-metrics with nonnegative integer weights in $G$ which realizes the distances within each hole. We develop a strongly polynomial purely combinatorial algorithm to find such a packing.
We prove that one can perfectly pack degenerate graphs into complete or dense $n$-vertex quasirandom graphs, provided that all the degenerate graphs have maximum degree $o(\frac{n}{\log n})$, and in addition $Ω(n)$ of them have at most $(1-Ω(1))n$ vertices and $Ω(n)$ leaves. This proves Ringel's conjecture and the Gyárfás Tree Packing Conjecture for all but an exponentially small fraction of trees (or sequences of trees, respectively).
A set $P$ of vertices in a graph $G$ is an open packing if no two distinct vertices in $P$ have a common neighbor. Among all maximal open packings in $G$, the smallest cardinality is denoted $ρ^{\rm o}_L(G)$ and the largest cardinality is $ρ^{\rm o}(G)$. There exist graphs for which these two invariants are arbitrarily far apart. In this paper we begin the investigation of the class of graphs that have one size of maximal open packings. By presenting a method of constructing such graphs we show that every graph is the induced subgraph of a graph in this class. The main result of the paper is a…
