Independence and matching numbers of some token graphs

Let $G$ be a graph of order $n$ and let $k\in\{1,\ldots,n-1\}$. The $k$-token graph $F_k(G)$ of $G$, is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is an edge of $G$. We study the independence and matching numbers of $F_k(G)$. We present a tight lower bound for the matching number of $F_k(G)$ for the case in which $G$ has either a perfect matching or an almost perfect matching. Also, we estimate the independence number for bipartite $k$-token graphs, and determine the exact value for some graphs.Comment: 16 pages, 4 figures. Third version is a major revision. Some proofs were corrected or simplified. New references adde

Independence and matching number for some token graphs

Let G be a graph of order n and let k ∈ {1, . . . , n−1}. The k-token graph Fk(G) of G is the graph whose vertices are the k-subsets of V (G), where two vertices are adjacent in Fk(G) whenever their symmetric difference is an edge of G. We study the independence and matching numbers of Fk(G). We present a tight lower bound for the matching number of Fk(G) for the case in which G has either a perfect matching or an almost perfect matching. Also, we estimate the independence number for bipartite ktoken graphs, and determine the exact value for some graphs

Five results on maximizing topological indices in graphs

In this paper, we prove a collection of results on graphical indices. We determine the extremal graphs attaining the maximal generalized Wiener index (e.g. the hyper-Wiener index) among all graphs with given matching number or independence number. This generalizes some work of Dankelmann, as well as some work of Chung. We also show alternative proofs for two recents results on maximizing the Wiener index and external Wiener index by deriving it from earlier results. We end with proving two conjectures. We prove that the maximum for the difference of the Wiener index and the eccentricity is attained by the path if the order $n$ is at least $9$ and that the maximum weighted Szeged index of graphs of given order is attained by the balanced complete bipartite graphs.Comment: 13 pages, 4 figure

A Note on the Sparing Number of Graphs

An integer additive set-indexer is defined as an injective function $f:V(G)\rightarrow 2^{\mathbb{N}_0}$ such that the induced function $g_f:E(G) \rightarrow 2^{\mathbb{N}_0}$ defined by $g_f (uv) = f(u)+ f(v)$ is also injective. An IASI $f$ is said to be a weak IASI if $|g_f(uv)|=max(|f(u)|,|f(v)|)$ for all $u,v\in V(G)$. A graph which admits a weak IASI may be called a weak IASI graph. The set-indexing number of an element of a graph $G$, a vertex or an edge, is the cardinality of its set-labels. The sparing number of a graph $G$ is the minimum number of edges with singleton set-labels, required for a graph $G$ to admit a weak IASI. In this paper, we study the sparing number of certain graphs and the relation of sparing number with some other parameters like matching number, chromatic number, covering number, independence number etc.Comment: 10 pages, 10 figures, submitte

A study of the total coloring of graphs.

The area of total coloring is a more recent and less studied area than vertex and edge coloring, but recently, some attention has been given to the Total Coloring Conjecture, which states that each graph\u27s total chromatic number xT is no greater than its maximum degree plus two. In this dissertation, it is proved that the conjecture is satisfied by those planar graphs in which no vertex of degree 5 or 6 1ies on more than three 3-cycles. The total independence number aT is found for some families of graphs, and a relationship between that parameter and the size of a graph\u27s minimum maximal matching is discussed. For colorings with natural numbers, the total chromatic sum ST is introduced, as is total strength (oT of a graph. Tools are developed for proving that a total coloring has minimum sum, and this sum is found for some graphs including paths, cycles, complete graphs, complete bipartite graphs, full binary trees, and some hypercubes. A family of graphs is found for which no optimal total coloring maximizes the smallest color class. Lastly, the relationship between a graph\u27s total chromatic number and its total strength is explored, and some graphs are found that require more than their total chromatic number of colors to obtain a minimum sum

KE Theory & the Number of Vertices Belonging to All Maximum Independent Sets in a Graph

For a graph $G$, let $\alpha (G)$ be the cardinality of a maximum independent set, let $\mu (G)$ be the cardinality of a maximum matching and let $\xi (G)$ be the number of vertices belonging to all maximum independent sets. Boros, Golumbic and Levit showed that in connected graphs where the independence number $\alpha (G)$ is greater than the matching number $\mu (G)$, $\xi (G) \geq 1 + \alpha(G) - \mu (G)$. For any graph $G$, we will show there is a distinguished induced subgraph $G[X]$ such that, under weaker assumptions, $\xi (G) \geq 1 + \alpha (G[X]) - \mu (G[X])$. Furthermore $1 + \alpha (G[X]) - \mu (G[X]) \geq 1 + \alpha (G) - \mu (G)$ and the difference between these bounds can be arbitrarily large. Lastly some results toward a characterization of graphs with equal independence and matching numbers is given

Large induced matchings in random graphs

Given a large graph $H$, does the binomial random graph $G(n,p)$ contain a copy of $H$ as an induced subgraph with high probability? This classical question has been studied extensively for various graphs $H$, going back to the study of the independence number of $G(n,p)$ by Erd\H{o}s and Bollob\'as, and Matula in 1976. In this paper we prove an asymptotically best possible result for induced matchings by showing that if $C/n\le p \le 0.99$ for some large constant $C$, then $G(n,p)$ contains an induced matching of order approximately $2\log_q(np)$, where $q= \frac{1}{1-p}$