Trees through specified vertices
نویسنده
چکیده
We prove a conjecture of Horak that can be thought of as an extension of classical results including Dirac’s theorem on the existence of Hamiltonian cycles. Namely, we prove for 1 ≤ k ≤ n − 2 if G is a connected graph with A ⊂ V (G) such that dG(v) ≥ k for all v ∈ A, then there exists a subtree T of G such that V (T ) ⊃ A and dT (v) ≤ ⌈ n−1 k ⌉ for all v ∈ A.
منابع مشابه
A NOTE ON THE EQUISEPARABLE TREES
Let T be a tree and n_{l}(eIT) and n_{2}(eIT) denote the number of vertices of T, lying on the two sides of the edge e. Suppose T_{l} and T_{2} are two trees with equal number of vertices, e in T_{1} and f in T_{2}. The edges e and f are said to be equiseparable if either n_{l}(eIT_{I}) = n_{l}(fIT_{2}) or n_{l}(eIT_{I}) = n_{2}(fIT_{2}). If there is an one-to-one correspondence between the ver...
متن کاملThe Generalised Randić Index of Trees
The Generalised Randić index R−α(T ) of a tree T is the sum over the edges uv of T of (d(u)d(v))−α where d(x) is the degree of the vertex x in T . For all α > 0, we find the minimal constant βc = βc(α) such that for all trees on at least 3 vertices R−α(T ) ≤ βc(n + 1) where n = |V (T )| is the number of vertices of T . For example, when α = 1, βc = 15 56 . This bound is sharp up to the additive...
متن کاملA Bijective Proof for the Number of Labeled q-Trees
We giv e a bijective proof that the number of vertex labeled q-trees on n vertices is given by n q [ q n − q + 1 ] n − q − 2 . The bijection transforms each pair ( S , f ) where S is a q-element subset of an n-set, and f is a function mapping an ( n − q − 2 )-set to a ( q n − q + 1 )-set into a labeled q-tree on n nodes by a cutand-paste process. As a special case, q = 1 yields a new bi...
متن کاملThe generalized Randic index of trees
The Generalised Randić index R−α(T ) of a tree T is the sum over the edges uv of T of (d(u)d(v))−α where d(x) is the degree of the vertex x in T . For all α > 0, we find the minimal constant βc = βc(α) such that for all trees on at least 3 vertices R−α(T ) ≤ βc(n + 1) where n = |V (T )| is the number of vertices of T . For example, when α = 1, βc = 15 56 . This bound is sharp up to the additive...
متن کاملOn the spectra of reduced distance matrix of the generalized Bethe trees
Let G be a simple connected graph and {v_1,v_2,..., v_k} be the set of pendent (vertices of degree one) vertices of G. The reduced distance matrix of G is a square matrix whose (i,j)-entry is the topological distance between v_i and v_j of G. In this paper, we compute the spectrum of the reduced distance matrix of the generalized Bethe trees.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Discrete Mathematics
دوره 309 شماره
صفحات -
تاریخ انتشار 2009