Interpolating between bounds on the independence number
نویسندگان
چکیده
For a non-negative integer T , we prove that the independence number of a graph G = (V,E) in which every vertex belongs to at most T triangles is at least ∑ u∈V f(d(u), T ) where d(u) denotes the degree of a vertex u ∈ V , f(d, T ) = 1 d+1 for T ≥ ( d 2 ) and f(d, T ) = (1 + (d2− d− 2T )f(d− 1, T ))/(d2 + 1− 2T ) for T < ( d 2 ) . This is a common generalization of the lower bounds for the independence number due to Caro, Wei, and Shearer. We discuss further possible strengthenings of our result and pose a corresponding conjecture.
منابع مشابه
Girth, minimum degree, independence, and broadcast independence
An independent broadcast on a connected graph $G$is a function $f:V(G)to mathbb{N}_0$such that, for every vertex $x$ of $G$, the value $f(x)$ is at most the eccentricity of $x$ in $G$,and $f(x)>0$ implies that $f(y)=0$ for every vertex $y$ of $G$ within distance at most $f(x)$ from $x$.The broadcast independence number $alpha_b(G)$ of $G$is the largest weight $sumlimits_{xin V(G)}f(x)$of an ind...
متن کاملCoverings, matchings and paired domination in fuzzy graphs using strong arcs
The concepts of covering and matching in fuzzy graphs using strong arcs are introduced and obtained the relationship between them analogous to Gallai’s results in graphs. The notion of paired domination in fuzzy graphs using strong arcs is also studied. The strong paired domination number γspr of complete fuzzy graph and complete bipartite fuzzy graph is determined and obtained bounds for the s...
متن کاملInterpolation Theorems on Graph Parameters
For each graph parameter f = f(G) of a graph G, it is said that f interpolates over the set of all graphs with a fixed degree sequence d,R(d) if the following statement holds: If m and M are the minimum and maximum values (respectively) of f(G) over all G ∈ R(d), then for any k, m ≤ k ≤ M, there is a graph G ∈ R(d) such that f(G) = k. In this paper, three graph parameters are shown to be interp...
متن کاملThe Independence Number Project: Α-bounds
A lower bound for the independence number of a graph is a graph invariant l such that, for every graph G, l(G) ≤ α(G). Similarly, an upper bound for the independence number is a graph invariant u such that, for every graph G, α(G) ≤ u(G). Many efficiently computable upper and lower bounds, called α-bounds here, have been published and these are surveyed in the following section. They can be use...
متن کاملLower bounds on the signed (total) $k$-domination number
Let $G$ be a graph with vertex set $V(G)$. For any integer $kge 1$, a signed (total) $k$-dominating functionis a function $f: V(G) rightarrow { -1, 1}$ satisfying $sum_{xin N[v]}f(x)ge k$ ($sum_{xin N(v)}f(x)ge k$)for every $vin V(G)$, where $N(v)$ is the neighborhood of $v$ and $N[v]=N(v)cup{v}$. The minimum of the values$sum_{vin V(G)}f(v)$, taken over all signed (total) $k$-dominating functi...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Discrete Mathematics
دوره 310 شماره
صفحات -
تاریخ انتشار 2010