Þ-energy of generalized Petersen graphs

Graph fall-coloring, also known as idomatic partition or independent domatic partition of graphs, was formally introduced by Dunbar, Hedetniemi, Hedetniemi, Jacobs, Knisely, Laskar and Rall in 2000 as an extension of both Graph Coloring and Graph Domination. It asks for a partition of the vertex set of a given graph into independent dominating sets, or equivalently into maximal independent sets. We study two fundamental questions related to this concept: when such a partition of vertices can exist, and how it relates to a proper coloring. We construct graphs with a large number of possible fall-colorings and as a consequence of that we answer a question of Dunbar et al. (J Combin Math Combin Comput 33:257–273, 2000) by constructing a family of graphs with arbitrarily far apart chromatic number and fall-chromatic number. In fact, we construct graphs whose Fall set (collection of k such that the graph has a fall coloring with k colors) is an arbitrarily long arithmetic sequence, thus giving us graphs with Fall set of large order having large gaps between its elements. We give a sufficient condition on the minimum degree for fall-colorable graphs and characterize the sharpness of this bound. Related to this, we also construct families of graphs which have both a k-fall-coloring and a k-coloring that is not a fall coloring for every k, illustrating the complex relationship between idomatic and non-domatic independent partitions.

Exact [formula omitted]-numbers of generalized Petersen graphs of certain higher-orders and on Möbius strips

A k‐weak bisection of a cubic graph G is a partition of the vertex‐set of G into two parts V1 and V2 of equal size, such that each connected component of the subgraph of G induced by () is a tree of at most vertices. This notion can be viewed as a relaxed version of nowhere‐zero flows, as it directly follows from old results of Jaeger that every cubic graph G with a circular nowhere‐zero r‐flow has a ‐weak bisection. In this article, we study problems related to the existence of k‐weak bisections. We believe that every cubic graph that has a perfect matching, other than the Petersen graph, admits a 4‐weak bisection and we present a family of cubic graphs with no perfect matching that do not admit such a bisection. The main result of this article is that every cubic graph admits a 5‐weak bisection. When restricted to bridgeless graphs, that result would be a consequence of the assertion of the 5‐flow Conjecture and as such it can be considered a (very small) step toward proving that assertion. However, the harder part of our proof focuses on graphs that do contain bridges.

How to build a brick

We offer a new, gradual approach to the largest girth problem for cubic graphs . It is easily observed that the largest possible girth of all ‐vertex cubic graphs is attained by a 2‐connected graph . By Petersen's graph theorem, is the disjoint union of a 2‐factor and a perfect matching . We refer to the edges of as ribs and classify the cycles in by their number of ribs. We define to be the smallest integer such that every cubic ‐vertex graph with a given perfect matching has a cycle of length at most with at most ribs. Here, we determine this function up to small additive constant for and up to a small multiplicative constant for larger .

We present a pseudo-deterministic NC algorithm for finding perfect matchings in bipartite graphs. Specifically, our algorithm is a randomized parallel algorithm which uses poly(n) processors, poly(log n) depth, poly(log n) random bits, and outputs for each bipartite input graph a unique perfect matching with high probability. That is, on the same graph it returns the same matching for almost all choices of randomness. As an immediate consequence we also find a pseudo-deterministic NC algorithm for constructing a depth first search (DFS) tree. We introduce a method for computing the union of all min-weight perfect matchings of a weighted graph in RNC and a novel set of weight assignments which in combination enable isolating a unique matching in a graph. We then show a way to use pseudo-deterministic algorithms to reduce the number of random bits used by general randomized algorithms. The main idea is that random bits can be reused by successive invocations of pseudo-deterministic randomized algorithms. We use the technique to show an RNC algorithm for constructing a depth first search (DFS) tree using only O(log^2 n) bits whereas the previous best randomized algorithm used O(log^7 n), and a new sequential randomized algorithm for the set-maxima problem which uses fewer random bits than the previous state of the art. Furthermore, we prove that resolving the decision question NC = RNC, would imply an NC algorithm for finding a bipartite perfect matching and finding a DFS tree in NC. This is not implied by previous randomized NC search algorithms for finding bipartite perfect matching, but is implied by the existence of a pseudo-deterministic NC search algorithm.

One-matching bi-Cayley graphs over abelian groups

On the 2-extendability of the generalized Petersen graphs

Hypergraphes de Petersen! Hypergraphes de Moore?

Let G be a simple undirected graph. The energy E(G) of G is the sum of the absolute values of the eigenvalues of the adjacent matrix of G, and the Hosoya index Z(G) of G is the total number of matchings in G. A tree is called a nonconjugated tree if it contains no perfect matching. Recently, Ou [‘Maximal Hosoya index and extremal acyclic molecular graphs without perfect matching’, Appl. Math. Lett.19 (2006), 652–656] determined the unique element which is maximal with respect to Z(G) among the family of nonconjugated n-vertex trees in the case of even n. In this paper, we provide a counterexample to Ou’s results. Then we determine the unique maximal element with respect to E(G) as well as Z(G) among the family of nonconjugated n-vertex trees for the case when n is even. As corollaries, we determine the maximal element with respect to E(G) as well as Z(G) among the family of nonconjugated chemical trees on n vertices, when n is even.

A bipartite graph $G(X,Y,E)$ with vertex partition $(X,Y)$ is said to have the Normalized Matching Property (NMP) if for any subset $S\subseteq X$ we have $\frac{|N(S)|}{|Y|}\geq\frac{|S|}{|X|}$. In this paper, we prove the following results about the Normalized Matching Property.
 
 The random bipartite graph $\mathbb{G}(k,n,p)$ with $|X|=k,|Y|=n$, and $k\leq n<\exp(k)$, and each pair $(x,y)\in X\times Y$ being an edge in $\mathbb{G}$ independently with probability $p$ has $p=\frac{\log n}{k}$ as the threshold for NMP. This generalizes a classic result of Erdős-Rényi on the $\frac{\log n}{n}$ threshold for the existence of a perfect matching in $\mathbb{G}(n,n,p)$.
 A bipartite graph $G(X,Y)$, with $k=|X|\le |Y|=n$, is said to be Thomason pseudorandom (following A. Thomason (Discrete Math., 1989)) with parameters $(p,\varepsilon)$ if every $x\in X$ has degree at least $pn$ and every pair of distinct $x, x'\in X$ have at most $(1+\varepsilon)p^2n$ common neighbours. We show that Thomason pseudorandom graphs have the following property: Given $\varepsilon>0$ and $n\geq k\gg 0$, there exist functions $f,g$ with $f(x), g(x)\to 0$ as $x\to 0$, and sets $\mathrm{Del}_X\subset X, \ \mathrm{Del}_Y\subset Y$ with $|\mathrm{Del}_X|\leq f(\varepsilon)k,\ |\mathrm{Del}_Y|\leq g(\varepsilon)n$ such that $G(X\setminus \mathrm{Del}_X,Y\setminus \mathrm{Del}_Y)$ has NMP. Enroute, we prove an 'almost' vertex decomposition theorem: Every Thomason pseudorandom bipartite graph $G(X,Y)$ admits - except for a negligible portion of its vertex set - a partition of its vertex set into graphs that are spanned by trees that have NMP, and which arise organically through the Euclidean GCD algorithm. 

This paper proposes an approach based on graph isomorphism to find the correspondence in relational matching. We describe a pseudo-automorphism group as Pseudo-aut (G) of a graph G, which is a set of all pseudo-automorphisms of G. We discuss some properties of the Pseudo-aut(C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ) and the relationships between various elements, establish the relationship between the pseudo-isomorphic and the perfect matching. From these we reach some important conclusions: the Petersen graph is a special element of the Pseudo-aut(C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sub> ); the composition of the Petersen graph is just one of its origins; there exists a Hamiltonian graph of order 12, which is 3-connected, 3-regular, non-planar, non-bipartite, and its girth is 5.

A subset S of vertices of a graph G is a dominating set of G if every vertex not in S has a neighbor in S, while S is a total dominating set of G if every vertex has a neighbor in S. If S is a dominating set with the additional property that the subgraph induced by S contains a perfect matching, then S is a paired-dominating set. The domination number, denoted γ(G), is the minimum cardinality of a dominating set of G, while the minimum cardinalities of a total dominating set and paired-dominating set are the total domination number, \gt(G), and the paired-domination number, \gp(G), respectively. For k ≥ 2, let G be a connected k-regular graph. It is known [Schaudt, Total domination versus paired domination, Discuss. Math. Graph Theory 32 (2012) 435--447] that \gpr(G)/γt(G) \le (2 k)/(k + 1). In the special case when k = 2, we observe that \gpr(G)/γt(G) \le 4/3, with equality if and only if G \cong C5. When k = 3, we show that \gpr(G)/γt(G) \le 3/2, with equality if and only if G is the Petersen graph. More generally for k ≥ 2, if G has girth at least 5 and satisfies \gpr(G)/γt(G) = (2 k)/(k + 1), then we show that G is a diameter-2 Moore graph. As a consequence of this result, we prove that for k ≥ 2 and k \ne 57, if G has girth at least 5, then \gpr(G)/γt(G) \le (2 k)/(k + 1), with equality if and only if k=2 and G \cong C5 or k = 3 and G is the Petersen graph.

A domatic partition of $$G$$ is the partition of vertices $$V\left( G \right)$$ into disjoint dominating sets. The maximum size of disjoint dominating sets is called the domatic number of $$G$$. In this paper, comparative results are investigated on domatic partition of few graphs of cycle related graphs such as complete graph, tadpole graph, lollipop graph and barbell graph for cognitive wireless sensor networks. Then the middle graph and central graph of these graphs are studied and domatic number of the defined graphs are determined to find the nodes in disjoint sets to disribute the tasks uniformly rather than burden the nodes in domatic set. Furthermore, diameter and domination number of these graphs are observed.

Exploits the close relationship between circular arc graphs and interval graphs to design efficient approximation algorithms for NP-hard optimization problems on circular arc graphs. The problems considered are maximum domatic partition and online minimum vertex coloring. We present a heuristic for the domatic partition problem with a performance ratio of 4. For online coloring, we consider two different online models. In the first model, arcs are presented in the increasing order of their left endpoints. For this model, our heuristic guarantees a solution which is within a factor of 2 of the optimal (off-line) value; and we show that no online coloring algorithm can achieve a performance guarantee of less than 3/2. In the second online model, arcs are presented in an arbitrary order; and it is known that no online coloring algorithm can achieve a performance guarantee of less than 3. For this model, we present a heuristic which provides a performance guarantee of 4. >