Super connectivity of Kronecker products of graphs

Let G₁ and G₂ be two connected graphs. The Kronecker product G₁×G₂ has vertex set V(G₁×G₂)=V(G₁)×V(G₂) and the edge set . In this paper, we show that if G is a bipartite graph with κ(G)=δ(G), then G×Kn(n?3) is super-κ.     

Minimum cycle bases of direct products of complete graphs

This paper presents a construction of a minimum cycle basis for the direct product of two complete graphs on three or more vertices. With the exception of two special cases, such bases consist entirely of triangles.     

On edge connectivity of direct products of graphs

Let λ(G) be the edge connectivity of G. The direct product of graphs G and H is the graph with vertex set V(G×H)=V(G)×V(H), where two vertices (u₁,v₁) and (u₂,v₂) are adjacent in G×H if u₁u₂∈E(G) and v₁v₂∈E(H). We prove that λ(G×Kn)=min{n(n−1)λ(G),(n−1)δ(G)} for every nontrivial graph G and n?3. We also prove that for almost every pair of graphs G and H with n vertices and edge probability p, G×H is k-connected, where k=O(²(n/logn)).     

Ranking numbers of graphs

Given a graph G, a vertex ranking (or simply, ranking) of G is a mapping f from V(G) to the set of all positive integers, such that for any path between two distinct vertices u and v with f(u)=f(v), there is a vertex w in the path with f(w)>f(u). If f is a ranking of G, the ranking number of G under f, denoted γf(G), is defined by , and the ranking number of G, denoted γ(G), is defined by . The vertex ranking problem is to determine the ranking number γ(G) of a given graph G. This problem is a natural model for the manufacturing scheduling problem. We study the ranking numbers of graphs in this paper. We consider the relation between the ranking numbers and the minimal cut sets, and the relation between the ranking numbers and the independent sets. From this, we obtain the ranking numbers of the powers of paths and the powers of cycles, the Cartesian product of P₂ with Pn or Cn, and the caterpilars. And we also find the vertex ranking numbers of the composition of two graphs in this paper.     

Vertex covering by paths on trees with its applications in machine translation

Given a tree and a set of paths in the tree, the problem of finding a minimum number of paths from the given path set to cover all the vertices in the tree is investigated in the paper. To distinguish from the classical path cover problem, such an optimization problem is referred to as vertex covering by paths. The problem and its edge variant, edge covering by paths, find applications in machine translation. Complexity and algorithmic results are presented for the problem and its edge variant.     

Domination number of Cartesian products of directed cycles

Let γ(G) denote the domination number of a digraph G and let Cm□Cn denote the Cartesian product of Cm and Cn, the directed cycles of length m,n?2. In Liu et al. (2010) [11], we determined the exact values of γ(Cm□Cn) when m=2,3,4. In this paper, we give a lower and upper bounds for γ(Cm□Cn). Furthermore, we prove a necessary and sufficient conditions for Cm□Cn to have an efficient dominating set. Also, we determine the exact values: γ(C₅□Cn)=2n; γ(C₆□Cn)=2n if n≡0(mod 3), otherwise, γ(C₆□Cn)=2n+2; if m≡0(mod 3) and n≡0(mod 3).     

A characterization of complete bipartite RAC graphs

A Right Angle Crossing drawing (RAC drawing for short) of a graph is such that edges can only cross at an angle of . In this paper we provide a characterization of the complete bipartite graphs that admit a straight-line RAC drawing.     

Orthogonal drawings and crossing numbers of the Kronecker product of two cycles

An orthogonal drawing of a graph is an embedding of the graph in the plane such that each edge is representable as a chain of alternately horizontal and vertical line segments. This style of drawing finds applications in areas such as optoelectronic systems, information visualization and VLSI circuits. We present orthogonal drawings of the Kronecker product of two cycles around vertex partitions of the graph into grids. In the process, we derive upper bounds on the crossing number of the graph. The resulting upper bounds are within a constant multiple of the lower bounds. Unlike the Cartesian product that is amenable to an inductive treatment, the Kronecker product entails a case-to-case analysis since the results depend heavily on the parameters corresponding to the lengths of the two cycles.     

On disjoint maximal independent sets in graphs

An old problem in graph theory is to characterize the graphs that admit two disjoint maximal independent sets.     

Connecting face hitting sets in planar graphs

We show that any face hitting set of size n of a connected planar graph with a minimum degree of at least 3 is contained in a connected subgraph of size 5n−6. Furthermore we show that this bound is tight by providing a lower bound in the form of a family of graphs. This improves the previously known upper and lower bound of 11n−18 and 3n respectively by Grigoriev and Sitters. Our proof is valid for simple graphs with loops and generalizes to graphs embedded in surfaces of arbitrary genus.     

Vertex rankings of chordal graphs and weighted trees

In this paper we consider the vertex ranking problem of weighted trees. We show that this problem is strongly NP-hard. We also give a polynomial-time reduction from the problem of vertex ranking of weighted trees to the vertex ranking of (simple) chordal graphs, which proves that the latter problem is NP-hard. In this way we solve an open problem of Aspvall and Heggernes. We use this reduction and the algorithm of Bodlaender et al.'s for vertex ranking of partial k-trees to give an exact polynomial-time algorithm for vertex ranking of a tree with bounded and integer valued weight functions. This algorithm serves as a procedure in designing a PTAS for weighted vertex ranking problem of trees with bounded weight functions.     

On graphs with the third largest number of maximal independent sets

Let G be a simple and undirected graph. By mi(G) we denote the number of maximal independent sets in G. Erd?s and Moser posed the problem to determine the maximum cardinality of mi(G) among all graphs of order n and to characterize the corresponding extremal graphs attaining this maximum cardinality. The above problem has been solved by Moon and Moser in [J.W. Moon, L. Moser, On cliques in graphs, Israel J. Math. 3 (1965) 23-28]. More recently, Jin and Li [Z. Jin, X. Li, Graphs with the second largest number of maximal independent sets, Discrete Mathematics 308 (2008) 5864-5870] investigated the second largest cardinality of mi(G) among all graphs of order n and characterized the extremal graph attaining this value of mi(G). In this paper, we shall determine the third largest cardinality of mi(G) among all graphs G of order n. Additionally, graphs achieving this value are also determined.     

Lucky labelings of graphs

Suppose the vertices of a graph G were labeled arbitrarily by positive integers, and let S(v) denote the sum of labels over all neighbors of vertex v. A labeling is lucky if the function S is a proper coloring of G, that is, if we have S(u)≠S(v) whenever u and v are adjacent. The least integer k for which a graph G has a lucky labeling from the set {1,2,…,k} is the lucky number of G, denoted by η(G).Using algebraic methods we prove that η(G)?k+1 for every bipartite graph G whose edges can be oriented so that the maximum out-degree of a vertex is at most k. In particular, we get that η(T)?2 for every tree T, and η(G)?3 for every bipartite planar graph G. By another technique we get a bound for the lucky number in terms of the acyclic chromatic number. This gives in particular that for every planar graph G. Nevertheless we offer a provocative conjecture that η(G)?χ(G) for every graph G.     

Perfect r-domination in the Kronecker product of two cycles, with an application to diagonal/toroidal mesh

If r?1, and m and n are each a multiple of (r+1)²+r², then each isomorphic component of C_m×C_n admits of a vertex partition into (r+1)²+r² perfect r-dominating sets. The result induces a dense packing of C_m×C_n by means of vertex-disjoint subgraphs, each isomorphic to a diagonal array. Areas of applications include efficient resource placement in a diagonal mesh and error-correcting codes.     

A 1.235 lower bound on the number of points needed to draw all n-vertex planar graphs

We study a problem of lower bounds on straight line drawings of planar graphs. We show that at least 1.235·n points in the plane are required to draw each n-vertex planar graph with edges drawn as straight line segments (for sufficiently large n). This improves the previous best bound of 1.206·n (for sufficiently large n) due to Chrobak and Karloff [Sigact News 20 (4) (1989) 83-86]. Our contribution is twofold: we improve the lower bound itself and we give a significantly simpler and more straightforward proof.     

The complexity for partitioning graphs by monochromatic trees,cycles and paths

Let G be an edge-coloured graph. We show in this paper that it is NP-hard to find the minimum number of vertex disjoint monochromatic trees which cover the vertices of the graph G. We also show that there is no constant factor approximation algorithm for the problem unless P?=?NP. The same results hold for the problem of finding the minimum number of vertex disjoint monochromatic cycles (paths, respectively) which cover the vertices of the graph.     

A note on 2-facial coloring of plane graphs

An l-facial coloring of a plane graph is a vertex coloring such that any two different vertices joined by a facial walk of length at most l receive distinct colors. It is known that every plane graph admits a 2-facial coloring using 8 colors [D. Král, T. Madaras, R. Škrekovski, Cyclic, diagonal and facial coloring, European J. Combin. 3-4 (26) (2005) 473-490]. We improve this bound for plane graphs with large girth and prove that if G is a plane graph with girth g?14 (resp. 10, 8) then G admits a 2-facial coloring using 5 colors (resp. 6, 7). Moreover, we give exact bounds for outerplanar graphs and K₄-minor free graphs.     

Oriented colorings of 2-outerplanar graphs

A graph G is 2-outerplanar if it has a planar embedding such that the subgraph obtained by removing the vertices of the outer face is outerplanar. The oriented chromatic number of an oriented graph H is defined as the minimum order of an oriented graph H^′ such that H has a homomorphism to H^′. In this paper, we prove that 2-outerplanar graphs are 4-degenerate. We also show that oriented 2-outerplanar graphs have a homomorphism to the Paley tournament QR₆₇, which implies that their (strong) oriented chromatic number is at most 67.