اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدPacking $(2^{k+1}-1)$-order perfect binary trees into (emph{k}+1)-connected graph
86
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Let $G=(V,E)$ and $H$ be two graphs. Packing problem is to find in $G$ the largest number of independent subgraphs each of which is isomorphic to $H$. Let $Usubset{V}$. If the graph $G-U$ has no subgraph isomorphic to $H$, $U$ is a cover of $G$. Covering problem is to find the smallest set $U$. The vertex-disjoint tree packing was not sufficiently discussed in literature but has its applications in data encryption and in communication networks such as multi-cast routing protocol design. In this paper, we give the kind of $(k+1)$-connected graph $G$ into which we can pack independently the subgraphs that are each isomorphic to the $(2^{k+1}-1)$-order perfect binary tree $T_k$. We prove that in $G$ the largest number of vertex-disjoint subgraphs isomorphic to $T_k$ is equal to the smallest number of vertices that cover all subgraphs isomorphic to $T_k$. Then, we propose that $T_k$ does not have the emph{ErdH{o}s-P{o}sa} property. We also prove that the $T_k$ packing problem in an arbitrary graph is NP-hard, and propose the distributed approximation algorithms.
قيم البحث
اقرأ أيضاً
The maximum size of an $r$-uniform hypergraph without a Berge cycle of length at least $k$ has been determined for all $k ge r+3$ by Furedi, Kostochka and Luo and for $k<r$ (and $k=r$, asymptotically) by Kostochka and Luo. In this paper, we settle th
e remaining cases: $k=r+1$ and $k=r+2$, proving a conjecture of Furedi, Kostochka and Luo.
In this paper, we give a type B analogue of the 1/k-Eulerian polynomials. Properties of this kind of polynomials, including combinatorial interpretations, recurrence relations and gamma-positivity are studied. In particular, we show that the 1/k-Eule
rian polynomials of type B are gamma-positive when $k>0$. Moreover, we obtain the corresponding results for derangements of type B. We show that a type B 1/k-derangement polynomials $d_n^B(x;k)$ are bi-gamma-positive when $kgeq 1/2$. In particular, we get a symmetric decomposition of $d_n^B(x;1/2)$ in terms of the classical derangement polynomials.
A $k$-sum of a set $Asubseteq mathbb{Z}$ is an integer that may be expressed as a sum of $k$ distinct elements of $A$. How large can the ratio of the number of $(k+1)$-sums to the number of $k$-sums be? Writing $kwedge A$ for the set of $k$-sums of $
A$ we prove that [ frac{|(k+1)wedge A|}{|kwedge A|}, le , frac{|A|-k}{k+1} ] whenever $|A|ge (k^{2}+7k)/2$. The inequality is tight -- the above ratio being attained when $A$ is a geometric progression. This answers a question of Ruzsa.
A defensive $k$-alliance in a graph is a set $S$ of vertices with the property that every vertex in $S$ has at least $k$ more neighbors in $S$ than it has outside of $S$. A defensive $k$-alliance $S$ is called global if it forms a dominating set. In
this paper we study the problem of partitioning the vertex set of a graph into (global) defensive $k$-alliances. The (global) defensive $k$-alliance partition number of a graph $Gamma=(V,E)$, ($psi_{k}^{gd}(Gamma)$) $psi_k^{d}(Gamma)$, is defined to be the maximum number of sets in a partition of $V$ such that each set is a (global) defensive $k$-alliance. We obtain tight bounds on $psi_k^{d}(Gamma)$ and $psi_{k}^{gd}(Gamma)$ in terms of several parameters of the graph including the order, size, maximum and minimum degree, the algebraic connectivity and the isoperimetric number. Moreover, we study the close relationships that exist among partitions of $Gamma_1times Gamma_2$ into (global) defensive $(k_1+k_2)$-alliances and partitions of $Gamma_i$ into (global) defensive $k_i$-alliances, $iin {1,2}$.
For a graph G=(V,E), the k-dominating graph of G, denoted by $D_{k}(G)$, has vertices corresponding to the dominating sets of G having cardinality at most k, where two vertices of $D_{k}(G)$ are adjacent if and only if the dominating set correspondin
g to one of the vertices can be obtained from the dominating set corresponding to the second vertex by the addition or deletion of a single vertex. We denote by $d_{0}(G)$ the smallest integer for which $D_{k}(G)$ is connected for all k greater than or equal to $d_{0}(G)$. It is known that $d_{0}(G)$ lies between $Gamma(G)+1$ and $|V|$ (inclusive), where ${Gamma}(G)$ is the upper domination number of G, but constructing a graph G such that $d_{0}(G)>{Gamma}(G)+1$ appears to be difficult. We present two related constructions. The first construction shows that for each integer k greater than or equal to 3 and each integer r from 1 to k-1, there exists a graph $G_{k,r}$ such that ${Gamma}(G_{k,r})=k, {gamma}(G_{k,r})=r+1$ and $d_{0}(G_{k,r})=k+r={Gamma}(G)+{gamma}(G)-1$. The second construction shows that for each integer k greater than or equal to 3 and each integer r from 1 to k-1, there exists a graph $Q_{k,r}$ such that ${Gamma}(Q_{k,r})=k, {gamma}(Q_{k,r})=r$ and $d_{0}(Q_{k,r})=k+r={Gamma}(G)+{gamma}(G)$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
