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Register a new userThe tree packing conjecture for trees of almost linear maximum degree
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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.
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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.
A subset of vertices is a {it maximum independent set} if no two of the vertices are adjacent and the subset has maximum cardinality. A subset of vertices is called a {it maximum dissociation set} if it induces a subgraph with vertex degree at most 1, and the subset has maximum cardinality. Zito [J. Graph Theory {bf 15} (1991) 207--221] proved that the maximum number of maximum independent sets of a tree of order $n$ is $2^{frac{n-3}{2}}$ if $n$ is odd, and $2^{frac{n-2}{2}}+1$ if $n$ is even and also characterized all extremal trees with the most maximum independent sets, which solved a question posed by Wilf. Inspired by the results of Zito, in this paper, by establishing four structure theorems and a result of $k$-K{o}nig-Egerv{a}ry graph, we show that the maximum number of maximum dissociation sets in a tree of order $n$ is begin{center} $left{ begin{array}{ll} 3^{frac{n}{3}-1}+frac{n}{3}+1, & hbox{if $nequiv0pmod{3}$;} 3^{frac{n-1}{3}-1}+1, & hbox{if $nequiv1pmod{3}$;} 3^{frac{n-2}{3}-1}, & hbox{if $nequiv2pmod{3}$,} end{array} right.$ end{center} and also give complete structural descriptions of all extremal trees on which these maxima are achieved.
Minimum $k$-Section denotes the NP-hard problem to partition the vertex set of a graph into $k$ sets of sizes as equal as possible while minimizing the cut width, which is the number of edges between these sets. When $k$ is an input parameter and $n$ denotes the number of vertices, it is NP-hard to approximate the width of a minimum $k$-section within a factor of $n^c$ for any $c<1$, even when restricted to trees with constant diameter. Here, we show that every tree $T$ allows a $k$-section of width at most $(k-1) (2 + 16n / diam(T) ) Delta(T)$. This implies a polynomial-time constant-factor approximation for the Minimum $k$-Section Problem when restricted to trees with linear diameter and constant maximum degree. Moreover, we extend our results from trees to arbitrary graphs with a given tree decomposition.
Loebl, Komlos, and Sos conjectured that any graph with at least half of its vertices of degree at least k contains every tree with at most k edges. We propose a version of this conjecture for skewed trees, i.e., we consider the class of trees with at most k edges such that the sizes of the colour classes of the trees have a given ratio. We show that our conjecture is asymptotically correct for dense graphs. The proof relies on the regularity method. Our result implies bounds on Ramsey number of several trees of given skew.
Given a simple graph $G$, denote by $Delta(G)$, $delta(G)$, and $chi(G)$ the maximum degree, the minimum degree, and the chromatic index of $G$, respectively. We say $G$ is emph{$Delta$-critical} if $chi(G)=Delta(G)+1$ and $chi(H)le Delta(G)$ for every proper subgraph $H$ of $G$; and $G$ is emph{overfull} if $|E(G)|>Delta lfloor |V(G)|/2 rfloor$. Since a maximum matching in $G$ can have size at most $lfloor |V(G)|/2 rfloor$, it follows that $chi(G) = Delta(G) +1$ if $G$ is overfull. Conversely, let $G$ be a $Delta$-critical graph. The well known overfull conjecture of Chetwynd and Hilton asserts that $G$ is overfull provided $Delta(G) > |V(G)|/3$. In this paper, we show that any $Delta$-critical graph $G$ is overfull if $Delta(G) - 7delta(G)/4ge(3|V(G)|-17)/4$.
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