Neighbor Connectivity of the Alternating Group Graph

Given a graph [Formula: see text], its neighbor connectivity is the least number of vertices whose deletion along with their neighbors results in a disconnected, complete, or empty graph. The edge neighbor connectivity is the least number of edges whose deletion along with their endpoints results in a disconnected, complete, or empty graph. In this paper, we determine the neighbor connectivity [Formula: see text] and the edge neighbor connectivity [Formula: see text] of the alternating group graph. We show that [Formula: see text], where [Formula: see text] is the [Formula: see text]-dimensional alternating group graph.

Families of Integral Cographs within a Triangular Array

The determinant Hosoya triangle, is a triangular array where the entries are the determinants of two-by-two Fibonacci matrices. The determinant Hosoya triangle mod 2 gives rise to three infinite families of graphs, that are formed by complete product (join) of (the union of) two complete graphs with an empty graph. We give a necessary and sufficient condition for a graph from these families to be integral.Some features of these graphs are: they are integral cographs, all graphs have at most five distinct eigenvalues, all graphs are either d-regular graphs with d =2, 4, 6, . . . or almost-regular graphs, and some of them are Laplacian integral. Finally we extend some of these results to the Hosoya triangle.

The Bipartite $K_{2,2}$-Free Process and Bipartite Ramsey Number $b(2, t)$

The bipartite Ramsey number $b(s,t)$ is the smallest integer $n$ such that every blue-red edge coloring of $K_{n,n}$ contains either a blue $K_{s,s}$ or a red $K_{t,t}$. In the bipartite $K_{2,2}$-free process, we begin with an empty graph on vertex set $X\cup Y$, $|X|=|Y|=n$. At each step, a random edge from $X\times Y$ is added under the restriction that no $K_{2,2}$ is formed. This step is repeated until no more edges can be added. In this note, we analyze this process and prove that the resulting graph shows that $b(2,t) =\Omega(t^{3/2}/\log t)$, thereby improving the best known lower bound.

On graphs with same metric and upper dimension

The metric representation of a vertex [Formula: see text] of a graph [Formula: see text] is a finite vector representing distances of [Formula: see text] with respect to vertices of some ordered subset [Formula: see text]. The set [Formula: see text] is called a minimal resolving set if no proper subset of [Formula: see text] gives distinct representations for all vertices of [Formula: see text]. The metric dimension of [Formula: see text] is the cardinality of the smallest (with respect to its cardinality) minimal resolving set and upper dimension is the cardinality of the largest minimal resolving set. We show the existence of graphs for which metric dimension equals upper dimension. We found an error in a result, defining the metric dimension of join of path and totally disconnected graph, of the paper by Shahida and Sunitha [On the metric dimension of join of a graph with empty graph ([Formula: see text]), Electron. Notes Discrete Math. 63 (2017) 435–445] and we give the correct form of the theorem and its proof.

Subdivision vertex & subdivision vertex neighborhood

Let us newly define the Subdivision vertex corona and subdivision vertex neighbourhood corona product of the Cycle graph $ C_m $, for any $ m\ge3 $ with empty graph $ \overline{K_n} $, for any $ n \ge1 $. Additionally we will give generalization result based on the graph $ S({C_m}\odot\overline{K_n})$ and $ S_N({C_m}\boxdot\overline{K_n})$ for any, $ m\ge 3, n\ge 1 $. Then we shall calculate the topological indices of the different types of Zagreb index for our newly defined graphs. The molecular graph of Subdivision vertex corona product of $ C_3 $ and $ \overline{K_2} $ i.e., $ S( C_3 \odot \overline{K_2} )$ can be represented in the form of tri-thioacetane.

Linear Operators That Preserve Two Genera of a Graph

If a graph can be embedded in a smooth orientable surface of genus g without edge crossings and can not be embedded on one of genus g − 1 without edge crossings, then we say that the graph has genus g. We consider a mapping on the set of graphs with m vertices into itself. The mapping is called a linear operator if it preserves a union of graphs and it also preserves the empty graph. On the set of graphs with m vertices, we consider and investigate those linear operators which map graphs of genus g to graphs of genus g and graphs of genus g + j to graphs of genus g + j for j ≤ g and m sufficiently large. We show that such linear operators are necessarily vertex permutations.

Eccentric Sequence of Graphs

The distance d(u, v) from a vertex u of graph G to a vertex v is the length of a shortest u to v path. The eccentric sequences were the first distance related sequences introduced for undirected graphs. The eccentricity e(v) of v is the distance of a farthest vertex from v. The eccentric sequence of a graph G is a list of the eccentricities of vertices of graph G arranged in non-decreasing order. In this paper we determine the eccentric sequence of join of an empty graph and path graph(ie fan graph) and the eccentric sequence of the Cartesian product of paths P2 and Pn (ie Ladder graph).

On the König deficiency of zero-reducible graphs

A confluent and terminating reduction system is introduced for graphs, which preserves the number of their perfect matchings. A union-find algorithm is presented to carry out reduction in almost linear time. The König property is investigated in the context of reduction by introducing the König deficiency of a graph G as the difference between the vertex covering number and the matching number of G. It is shown that the problem of finding the König deficiency of a graph is NP-complete even if we know that the graph reduces to the empty graph. Finally, the König deficiency of graphs G having a vertex v such that $$G-v$$G-v has a unique perfect matching is studied in connection with reduction.

Saturation Games for Odd Cycles

Given a family of graphs $\mathcal{F}$, we consider the $\mathcal{F}$-saturation game. In this game, two players alternate adding edges to an initially empty graph on $n$ vertices, with the only constraint being that neither player can add an edge that creates a subgraph that lies in $\mathcal{F}$. The game ends when no more edges can be added to the graph. One of the players wishes to end the game as quickly as possible, while the other wishes to prolong the game. We let $\textrm{sat}_g(\mathcal{F};n)$ denote the number of edges that are in the final graph when both players play optimally. The $\{C_3\}$-saturation game was the first saturation game to be considered, but as of now the order of magnitude of $\textrm{sat}_g(\{C_3\},n)$ remains unknown. We consider a variation of this game. Let $\mathcal{C}_{2k+1}:=\{C_3,\ C_5,\ldots,C_{2k+1}\}$. We prove that $\textrm{sat}_g(\mathcal{C}_{2k+1};n)\ge(\frac{1}{4}-\epsilon_k)n^2+o(n^2)$ for all $k\ge 2$ and that $\textrm{sat}_g(\mathcal{C}_{2k+1};n)\le (\frac{1}{4}-\epsilon'_k)n^2+o(n^2)$ for all $k\ge 4$, with $\epsilon_k<\frac{1}{4}$ and $\epsilon'_k>0$ constants tending to 0 as $k\to \infty$. In addition to this we prove $\textrm{sat}_g(\{C_{2k+1}\};n)\le \frac{4}{27}n^2+o(n^2)$ for all $k\ge 2$, and $\textrm{sat}_g(\mathcal{C}_\infty\setminus C_3;n)\le 2n-2$, where $\mathcal{C}_\infty$ denotes the set of all odd cycles.

Clumsy Packings of Graphs

Let $G$ and $H$ be graphs. We say that $P$ is an $H$-packing of $G$ if $P$ is a set of edge-disjoint copies of $H$ in $G$. An $H$-packing $P$ is maximal if there is no other $H$-packing of $G$ that properly contains P. Packings of maximum cardinality have been studied intensively, with several recent breakthrough results. Here, we consider minimum cardinality maximal packings. An $H$-packing $P$ is called clumsy if it is maximal of minimum size. Let $\mathrm{cl}(G,H)$ be the size of a clumsy $H$-packing of $G$. We provide nontrivial bounds for $\mathrm{cl}(G,H)$, and in many cases asymptotically determine $\mathrm{cl}(G,H)$ for some generic classes of graphs G such as $K_n$ (the complete graph), $Q_n$ (the cube graph), as well as square, triangular, and hexagonal grids. We asymptotically determine $\mathrm{cl}(K_n,H)$ for every fixed non-empty graph $H$. In particular, we prove that $$\mathrm{cl}(K_n, H) = \frac{\binom{n}{2}- \mathrm{ex}(n,H)}{|E(H)|}+o(\mathrm{ex}(n,H)),$$where $ex(n,H)$ is the extremal number of $H$. A related natural parameter is $\mathrm{cov}(G,H)$, that is the smallest number of copies of $H$ in $G$ (not necessarily edge-disjoint) whose removal from $G$ results in an $H$-free graph. While clearly $\mathrm{cov}(G,H) \leqslant\mathrm{cl}(G,H)$, all of our lower bounds for $\mathrm{cl}(G,H)$ apply to $\mathrm{cov}(G,H)$ as well.