A class of self-complementary vertex-transitive digraphs

We characterize the class of self-complementary vertex-transitive digraphs on a prime number p of vertices. Using this, we enumerate (i) self-complementary strongly vertex-transitive digraphs on p vertices, (ii) self-complementary vertex-transitive digraphs on p vertices, (iii) self-complementary vertex-transitive graphs on p vertices. Finally it is shown that every self-complementary vertex-transitive digraph on p vertices is either a tournament or a graph.     

On the construction and enumeration of minimal blocks

In this paper a method is given for constructing and enumerating the minimal blocks on a given number of vertices using the graphs on a smaller number of vertices. Minimal blocks having up to 12 vertices have been constructed in this way.     

Chromaticity of two-trees

The graphs called 2-trees are defined by recursion. The smallest 2-tree is the complete graph on 2 vertices. A 2-tree on n + 1 vertices (where n ≥ 2) is obtained by adding a new vertex adjacent to each of 2 arbitrarily selected adjacent vertices in a 2-tree on n vertices. A graph G is a 2-tree on n(≥2) vertices if and only if its chromatic polynomial is equal to γ(γ - 1)(γ - 2)^n—2.     

Local structure of 5- and 6-connected graphs

We prove that if graph on n vertices is minimally and contraction critically 5-connected, then it has 4n/7 vertices of degree 5. We also prove that if graph on n vertices is minimally and contraction critically 6-connected, then it has n/2 vertices of degree 6. Bibliography: 7 titles.     

An involution on increasing trees

We introduce a method to construct bijections on increasing trees. Using this method, we construct an involution on increasing trees, from which we obtain the equidistribution of the statistics ‘number of odd vertices’ and ‘number of even vertices at odd levels’. As an application, we deduce that the expected value of the number of even vertices is twice the expected value of the number of odd vertices in a random recursive tree of given size.     

Master corner polyhedron: Vertices

We focus on the vertices of the master corner polyhedron (MCP), a fundamental object in the theory of integer linear programming. We introduce two combinatorial operations that transform vertices to their neighbors. This implies that each MCP can be defined by the initial vertices regarding these operations; we call them support vertices. We prove that the class of support vertices of all MCPs over a group is invariant under automorphisms of this group and describe MCP vertex bases. Among other results, we characterize its irreducible points, establish relations between a vertex and the nontrivial facets that pass through it, and prove that this polyhedron is of diameter 2.     

Vertex-faithful regular polyhedra

We study the abstract regular polyhedra with automorphism groups that act faithfully on their vertices, and show that each non-flat abstract regular polyhedron covers a “vertex-faithful” polyhedron with the same number of vertices. We then use this result and earlier work on flat polyhedra to study abstract regular polyhedra based on the size of their vertex set. In particular, we classify all regular polyhedra where the number of vertices is prime or twice a prime. We also construct the smallest regular polyhedra with a prime squared number of vertices.     

A graph for which the inertia bound is not tight

The inertia bound gives an upper bound on the independence number of a graph by considering the inertia of matrices corresponding to the graph. The bound is known to be tight for graphs on 10 or fewer vertices as well as for all perfect graphs. It is natural to question whether the bound is always tight. We show that the bound is not tight for the Paley graph on 17 vertices as well as its induced subgraph on 16 vertices.     

A graph which is edge transitive but not arc transitive

A graph having 27 vertices is described, whose automorphism group is transitive on vertices and undirected edges, but not on directed edges.     

Complexity and exact algorithms for vertex multicut in interval and bounded treewidth graphs

Multicut is a fundamental network communication and connectivity problem. It is defined as: given an undirected graph and a collection of pairs of terminal vertices, find a minimum set of edges or vertices whose removal disconnects each pair. We mainly focus on the case of removing vertices, where we distinguish between allowing or disallowing the removal of terminal vertices. Complementing and refining previous results from the literature, we provide several NP-completeness and (fixed-parameter) tractability results for restricted classes of graphs such as trees, interval graphs, and graphs of bounded treewidth.     

Inflection points and affine vertices of closed curves on 2-dimensional affine flat tori

This paper deals with a geometric problem on inflection points and affine vertices for closed curves in an affine flat torus. We show that the least number of inflection points lying on a closed curve that is not homotopic to zero is 2 if the torus is affinely equivalent to a euclidean torus and 0 otherwise. We consider also the number of affine vertices on a strictly convex closed curve on a flat torus. An explicit example of a closed curve with six affine vertices is given.     

Small embeddings for partial cycle systems of odd length

We prove that if m is odd then a partial m-cycle system on n vertices can be embedded in an m-cycle system on at most m((m − 2)n(n − 1) + 2n + 1) vertices and that a partial weak Steiner m-cycle system on n vertices can be embedded in an m-cycle system on m(2n + 1) vertices.     

Minimal Ellipsoid Circumscribing a Polytope Defined by a System of Linear Inequalities

In this paper, we will propose algorithms for calculating a minimal ellipsoid circumscribing a polytope defined by a system of linear inequalities. If we know all vertices of the polytope and its cardinality is not very large, we can solve the problem in an efficient manner by a number of existent algorithms. However, when the polytope is defined by linear inequalities, these algorithms may not work since the cardinality of vertices may be huge. Based on a fact that vertices determining an ellipsoid are only a fraction of these vertices, we propose algorithms which iteratively calculate an ellipsoid which covers a subset of vertices. Numerical experiment shows that these algorithms perform well for polytopes of dimension up to seven.     

Intervals in matroid basis graphs

An interval in a graph is a subgraph induced by all the vertices on shortest paths between two given vertices. Intervals in matroid basis graphs satisfy many nice properties. Key results are: (1) any two vertices of a basis graph are together in some longest interval; (2) every basis graph with the minimum number of vertices for its diameter is an interval, indeed a hypercube. (1) turns out to be a simple case of a theorem in Edmonds' theory of matroid partition.     

The (signless) Laplacian spectral radius of unicyclic and bicyclic graphs with <Emphasis Type="Italic">n</Emphasis> vertices and <Emphasis Type="Italic">k</Emphasis> pendant vertices

In this paper, the effects on the signless Laplacian spectral radius of a graph are studied when some operations, such as edge moving, edge subdividing, are applied to the graph. Moreover, the largest signless Laplacian spectral radius among the all unicyclic graphs with n vertices and k pendant vertices is identified. Furthermore, we determine the graphs with the largest Laplacian spectral radii among the all unicyclic graphs and bicyclic graphs with n vertices and k pendant vertices, respectively.     

Graphs with unique maximal clumpings

A partition of the vertices of a graph is called a clumping if, for vertices in distinct partition classes, adjacency depends only on the partition classes, not on the specific vertices. We give a simple necessary and sufficient condition for a finite graph to have a unique maximal clumping. We also investigate the extent to which this and related results generalize to infinite graphs.     

A solitaire game played on 2-colored graphs

We introduce a solitaire game played on a graph. Initially one disk is placed at each vertex, one face green and the other red, oriented with either color facing up. Each move of the game consists of selecting a vertex whose disk shows green, flipping over the disks at neighboring vertices, and deleting the selected vertex. The game is won if all vertices are eliminated. We derive a simple parity-based necessary condition for winnability of a given game instance. By studying graph operations that construct new graphs from old ones, we obtain broad classes of graphs where this condition also suffices, thus characterizing the winnable games on such graphs. Concerning two familiar (but narrow) classes of graphs, we show that for trees a game is winnable if and only if the number of green vertices is odd, and for n-cubes a game is winnable if and only if the number of green vertices is even and not all vertices have the same color. We provide a linear-time algorithm for deciding winnability for games on maximal outerplanar graphs. We reduce the decision problem for winnability of a game on an arbitrary graph G to winnability of games on its blocks, and to winnability on homeomorphic images of G obtained by contracting edges at 2-valent vertices.     

Graphs with equal eternal vertex cover and eternal domination numbers

Mobile guards on the vertices of a graph are used to defend it against an infinite sequence of attacks on either its vertices or its edges. If attacks occur at vertices, this is known at the eternal domination problem. If attacks occur at edges, this is known as the eternal vertex cover problem. We focus on the model in which all guards can move to neighboring vertices in response to an attack. Motivated by the question of which graphs have equal eternal vertex cover and eternal domination numbers, a number of results are presented; one of the main results of the paper is that the eternal vertex cover number is greater than the eternal domination number (in the all-guards move model) in all graphs of minimum degree at least two.     

Colorings of plane graphs: A survey

After a brief historical account, a few simple structural theorems about plane graphs useful for coloring are stated, and two simple applications of discharging are given. Afterwards, the following types of proper colorings of plane graphs are discussed, both in their classical and choosability (list coloring) versions: simultaneous colorings of vertices, edges, and faces (in all possible combinations, including total coloring), edge-coloring, cyclic coloring (all vertices in any small face have different colors), 3-coloring, acyclic coloring (no 2-colored cycles), oriented coloring (homomorphism of directed graphs to small tournaments), a special case of circular coloring (the colors are points of a small cycle, and the colors of any two adjacent vertices must be nearly opposite on this cycle), 2-distance coloring (no 2-colored paths on three vertices), and star coloring (no 2-colored paths on four vertices). The only improper coloring discussed is injective coloring (any two vertices having a common neighbor should have distinct colors).     

A lower bound for the independence number of a planar graph

A subset of the vertices of a graph is independent if no two vertices in the subset are adjacent. The independence number α(G) is the maximum number of vertices in an independent set. We prove that if G is a planar graph on N vertices then $\text{α(G)/N ?}\text{2}\text{9}$.