Some topological properties of star graphs: The surface area and volume

The star graph, as an interesting network topology, has been extensively studied in the past. In this paper, we address some of the combinatorial properties of the star graph. In particular, we consider the problem of calculating the surface area and volume of the star graph, and thus answering an open problem previously posed in the literature. The surface area of a sphere with radius i in a graph is the number of nodes in the graph whose distance from a given node is exactly i. The volume of a sphere with radius i in a graph is the number of nodes within distance i from the given node. In this paper, we derive explicit expressions to calculate the surface area and volume in the star graph.     

Mutually independent hamiltonian cycles for the pancake graphs and the star graphs

A hamiltonian cycle C of a graph G is an ordered set u₁,u₂,…,u_n(G),u₁ of vertices such that u_i≠u_j for i≠j and u_i is adjacent to u_i+1 for every i{1,2,…,n(G)−1} and u_n(G) is adjacent to u₁, where n(G) is the order of G. The vertex u₁ is the starting vertex and u_i is the ith vertex of C. Two hamiltonian cycles C₁=u₁,u₂,…,u_n(G),u₁ and C₂=v₁,v₂,…,v_n(G),v₁ of G are independent if u₁=v₁ and u_i≠v_i for every i{2,3,…,n(G)}. A set of hamiltonian cycles {C₁,C₂,…,C_k} of G is mutually independent if its elements are pairwise independent. The mutually independent hamiltonicity IHC(G) of a graph G is the maximum integer k such that for any vertex u of G there exist k mutually independent hamiltonian cycles of G starting at u.In this paper, the mutually independent hamiltonicity is considered for two families of Cayley graphs, the n-dimensional pancake graphs P_n and the n-dimensional star graphs S_n. It is proven that IHC(P₃)=1, IHC(P_n)=n−1 if n≥4, IHC(S_n)=n−2 if n{3,4} and IHC(S_n)=n−1 if n≥5.     

Global defensive alliances in star graphs

A defensive alliance in a graph G=(V,E) is a set of vertices S⊆V satisfying the condition that, for each v∈S, at least one half of its closed neighbors are in S. A defensive alliance S is called a critical defensive alliance if any vertex is removed from S, then the resulting vertex set is not a defensive alliance any more. An alliance S is called global if every vertex in V(G)?S is adjacent to at least one member of the alliance S. In this paper, we shall propose a way for finding a critical global defensive alliance of star graphs. After counting the number of vertices in the critical global defensive alliance, we can derive an upper bound to the size of the minimum global defensive alliances in star graphs.     

List star chromatic index of sparse graphs

A star edge-coloring of a graph $G$ is a proper edge coloring such that every 2-colored connected subgraph of $G$ is a path of length at most 3. For a graph $G$, let the list star chromatic index of $G$, $c{h}_s^{\prime }\left(G\right)$, be the minimum $k$ such that for any $k$-uniform list assignment $L$ for the set of edges, $G$ has a star edge-coloring from $L$. Dvo?ák et al. (2013) asked whether the list star chromatic index of every subcubic graph is at most 7. In Kerdjoudj et al. (2017) we proved that it is at most 8. In this paper we consider graphs with any maximum degree, we proved that if the maximum average degree of a graph $G$ is less than $\frac{14}{5}$ (resp. 3), then $c{h}_s^{\prime }\left(G\right)\le 2\Delta \left(G\right)+2$ (resp. $c{h}_s^{\prime }\left(G\right)\le 2\Delta \left(G\right)+3$).     

Connectivity of Cartesian product graphs

Use v_i,κ_i,λ_i,δ_i to denote order, connectivity, edge-connectivity and minimum degree of a graph G_i for i=1,2, respectively. For the connectivity and the edge-connectivity of the Cartesian product graph, up to now, the best results are κ(G₁×G₂)?κ₁+κ₂ and λ(G₁×G₂)?λ₁+λ₂. This paper improves these results by proving that κ(G₁×G₂)?min{κ₁+δ₂,κ₂+δ₁} and λ(G₁×G₂)=min{δ₁+δ₂,λ₁v₂,λ₂v₁} if G₁ and G₂ are connected undirected graphs; κ(G₁×G₂)?min{κ₁+δ₂,κ₂+δ₁,2κ₁+κ₂,2κ₂+κ₁} if G₁ and G₂ are strongly connected digraphs. These results are also generalized to the Cartesian products of connected graphs and n strongly connected digraphs, respectively.     

Color-critical graphs have logarithmic circumference

A graph G is k-critical if every proper subgraph of G is (k−1)-colorable, but the graph G itself is not. We prove that every k-critical graph on n vertices has a cycle of length at least , improving a bound of Alon, Krivelevich and Seymour from 2000. Examples of Gallai from 1963 show that the bound cannot be improved to exceed . We thus settle the problem of bounding the minimal circumference of k-critical graphs, raised by Dirac in 1952 and Kelly and Kelly in 1954.     

Embedding cycles of various lengths into star graphs with both edge and vertex faults

The n-dimensional star graph S_n is an attractive alternative to the hypercube graph and is a bipartite graph with two partite sets of equal size. Let F_v and F_e be the sets of faulty vertices and faulty edges of S_n, respectively. We prove that S_n − F_v − F_e contains a fault-free cycle of every even length from 6 to n! − 2∣F_v∣ with ∣F_v∣ + ∣F_e∣ ? n − 3 for every n ? 4. We also show that S_n − F_v − F_e contains a fault-free path of length n! − 2∣F_v∣ − 1 (respectively, n! − 2∣F_v∣ − 2) between two arbitrary vertices of S_n in different partite sets (respectively, the same partite set) with ∣F_v∣ + ∣F_e∣ ? n − 3 for every n ? 4.     

Containment properties of product and power graphs

In this paper, we study containment properties of graphs in relation with the Cartesian product operation. These results can be used to derive embedding results for interconnection networks for parallel architectures.First, we show that the isomorphism of two Cartesian powers G^r and H^r implies the isomorphism of G and H, while G^r⊆H^r does not imply G⊆H, even for the special cases when G and H are prime, and when they are connected and have the same number of nodes at the same time.Then, we find a simple sufficient condition under which the containment of products implies the containment of the factors: if , where all graphs G_i are connected and no graph H_j has 4-cycles, then each G_i is a subgraph of a different graph H_j. Hence, if G is connected and H has no 4-cycles, then G^r⊆H^r implies G⊆H.Finally, we focus on the particular case of products of graphs with the linear array. We show that the fact that G×L_n⊆H×L_n does not imply that G⊆H even in the case when G and H are connected and have the same number of nodes. However, we find a sufficient condition under which G×L_n⊆H×L_n implies G⊆H.     

Classes of graphs without star forests and related graphs

This work provides a structural characterisation of hereditary graph classes that do not contain a star forest, several graphs obtained from star forests by subset complementation, a union of cliques, and the complement of a union of cliques as induced subgraphs. This provides, for instance, structural results for graph classes not containing a matching and several complements of a matching. In terms of the speed of hereditary graph classes, our results imply that all such classes have at most factorial speed of growth.     

Some results on R 2-edge-connectivity of even regular graphs

Let G be a connected k(≥3)-regular graph with girth g. A set S of the edges in G is called an Rredge-cut if G-S is disconnected and comains neither an isolated vertex nor a one-degree vertex. The R2-edge-connectivity of G, denoted by λ^n(G), is the minimum cardinality over all R2-edge-cuts, which is an important measure for fault-tolerance of computer interconnection networks. In this paper, λ^n(G)=g(2k-2) for any 2k-regular connected graph G (≠K5) that is either edge-transitive or vertex-transitive and g≥5 is given.     

On the diameter of Kronecker graphs

We prove that a.a.s. as soon as a Kronecker graph becomes connected it has a finite diameter.     

Bipartite graphs without a skew star

A skew star is a tree with exactly three vertices of degree one being at distance 1, 2, 3 from the only vertex of degree three. In the present paper, we propose a structural characterization for the class of bipartite graphs containing no skew star as an induced subgraph and discuss some applications of the obtained result.     

Edge-forwarding index of star graphs and other Cayley graphs

We present a technique for building, in some Cayley graphs, a routing for which the load of every edge is almost the same. This technique enables us to find the edge-forwarding index of star graphs and complete-transposition graphs.     

List star edge coloring of k-degenerate graphs

A star edge coloring of a graph is a proper edge coloring such that every connected 2-colored subgraph is a path with at most 3 edges. Deng et al. and Bezegová et al. independently show that the star chromatic index of a tree with maximum degree $\Delta$ is at most $?\frac{3\Delta }{2}?$, which is tight. In this paper, we study the list star edge coloring of $k$-degenerate graphs. Let $c{h}_{st}^{\prime }\left(G\right)$ be the list star chromatic index of $G$: the minimum $s$ such that for every $s$-list assignment $L$ for the edges, $G$ has a star edge coloring from $L$. By introducing a stronger coloring, we show with a very concise proof that the upper bound on the star chromatic index of trees also holds for list star chromatic index of trees, i.e. $c{h}_{st}^{\prime }\left(T\right)\le ?\frac{3\Delta }{2}?$ for any tree $T$ with maximum degree $\Delta$. And then by applying some orientation technique we present two upper bounds for list star chromatic index of $k$-degenerate graphs.     

Stochastic kronecker graphs

A random graph model based on Kronecker products of probability matrices has been recently proposed as a generative model for large‐scale real‐world networks such as the web. This model simultaneously captures several well‐known properties of real‐world networks; in particular, it gives rise to a heavy‐tailed degree distribution, has a low diameter, and obeys the densification power law. Most properties of Kronecker products of graphs (such as connectivity and diameter) are only rigorously analyzed in the deterministic case. In this article, we study the basic properties of stochastic Kronecker products based on an initiator matrix of size two (which is the case that is shown to provide the best fit to many real‐world networks). We will show a phase transition for the emergence of the giant component and another phase transition for connectivity, and prove that such graphs have constant diameters beyond the connectivity threshold, but are not searchable using a decentralized algorithm. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 453–466, 2011     

Strong list-chromatic index of subcubic graphs

A strong $k$-edge-coloring of a graph G is an edge-coloring with $k$ colors in which every color class is an induced matching. The strong chromatic index of $G$, denoted by ${\chi }_s^{\prime }\left(G\right)$, is the minimum $k$ for which $G$ has a strong $k$-edge-coloring. In 1985, Erd?s and Ne?et?il conjectured that ${\chi }_s^{\prime }\left(G\right)\le \frac{5}{4}\Delta {\left(G\right)}^2$, where $\Delta \left(G\right)$ is the maximum degree of $G$. When $G$ is a graph with maximum degree at most 3, the conjecture was verified independently by Andersen and Horák, Qing, and Trotter. In this paper, we consider the list version of strong edge-coloring. In particular, we show that every subcubic graph has strong list-chromatic index at most 11 and every planar subcubic graph has strong list-chromatic index at most 10.