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We consider numbers and sizes of independent sets in graphs with minimum degree at least $d$, when the number $n$ of vertices is large. In particular we investigate which of these graphs yield the maximum numbers of independent sets of different sizes, and which yield the largest random independent sets. We establish a strengthened form of a conjecture of Galvin concerning the first of these topics.
Given a graph $G$, let $f_{G}(n,m)$ be the minimal number $k$ such that every $k$ independent $n$-sets in $G$ have a rainbow $m$-set. Let $mathcal{D}(2)$ be the family of all graphs with maximum degree at most two. Aharoni et al. (2019) conjectured t
We determine the computational complexity of approximately counting and sampling independent sets of a given size in bounded-degree graphs. That is, we identify a critical density $alpha_c(Delta)$ and provide (i) for $alpha < alpha_c(Delta)$ randomiz
The degree-based entropy of a graph is defined as the Shannon entropy based on the information functional that associates the vertices of the graph with the corresponding degrees. In this paper, we study extremal problems of finding the graphs attain
Let $G$ be a simple graph with maximum degree $Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>Delta(G)lfloor |V(H)|/2 rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ with $Delta(G)>|V(G)|/3$ has chromatic index $Delta(G)$ i
The weight of a subgraph $H$ in $G$ is the sum of the degrees in $G$ of vertices of $H$. The {em height} of a subgraph $H$ in $G$ is the maximum degree of vertices of $H$ in $G$. A star in a given graph is minor if its center has degree at most five