اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدDecomposing highly edge-connected graphs into paths of any given length
88
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In 2006, Barat and Thomassen posed the following conjecture: for each tree $T$, there exists a natural number $k_T$ such that, if $G$ is a $k_T$-edge-connected graph and $|E(G)|$ is divisible by $|E(T)|$, then $G$ admits a decomposition into copies of $T$. This conjecture was verified for stars, some bistars, paths of length $3$, $5$, and $2^r$ for every positive integer $r$. We prove that this conjecture holds for paths of any fixed length.
قيم البحث
اقرأ أيضاً
We study the Decomposition Conjecture posed by Barat and Thomassen (2006), which states that for every tree $T$ there exists a natural number $k_T$ such that, if $G$ is a $k_T$-edge-connected graph and $|E(T)|$ divides $|E(G)|$, then $G$ admits a dec
omposition into copies of $T$. In a series of papers, Thomassen verified this conjecture for stars, some bistars, paths of length $3$, and paths whose length is a power of $2$. We verify the Decomposition Conjecture for paths of length $5$.
We consider a generalisation of Kellys conjecture which is due to Alspach, Mason, and Pullman from 1976. Kellys conjecture states that every regular tournament has an edge decomposition into Hamilton cycles, and this was proved by Kuhn and Osthus for
large tournaments. The conjecture of Alspach, Mason, and Pullman asks for the minimum number of paths needed in a path decomposition of a general tournament $T$. There is a natural lower bound for this number in terms of the degree sequence of $T$ and it is conjectured that this bound is correct for tournaments of even order. Almost all cases of the conjecture are open and we prove many of them.
A graph is locally irregular if any pair of adjacent vertices have distinct degrees. A locally irregular decomposition of a graph $G$ is a decomposition $mathcal{D}$ of $G$ such that every subgraph $H in mathcal{D}$ is locally irregular. A graph is s
aid to be decomposable if it admits a locally irregular decomposition. We prove that any decomposable split graph can be decomposed into at most three locally irregular subgraphs and we characterize all split graphs whose decomposition can be into one, two or three locally irregular subgraphs.
Given a graph $G$, a decomposition of $G$ is a partition of its edges. A graph is $(d, h)$-decomposable if its edge set can be partitioned into a $d$-degenerate graph and a graph with maximum degree at most $h$. For $d le 4$, we are interested in the
minimum integer $h_d$ such that every planar graph is $(d,h_d)$-decomposable. It was known that $h_3 le 4$, $h_2le 8$, and $h_1 = infty$. This paper proves that $h_4=1, h_3=2$, and $4 le h_2 le 6$.
We prove that for $k in mathbb{N}$ and $d leq 2k+2$, if a graph has maximum average degree at most $2k + frac{2d}{d+k+1}$, then $G$ decomposes into $k+1$ pseudoforests, where one of the pseudoforests has all connected components having at most $d$ edges.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
