Block transitivity and degree matrices: (Extended abstract)

We say that a square matrix M is a degree matrix of a given graph G if there is a so called equitable partition of its vertices into r blocks such that whenever two vertices belong to the same block, they have the same number of neighbors inside any block.We ask now whether for a given degree matrix M, there exists a graph G such that M is a degree matrix of G, and in addition, for any two edges e, f spanning between the same pair of blocks there exists an automorphism of G that sends e to f. In this work, we fully characterize the matrices for which such a graph exists and show a way to construct one.     

On defect-d matchings in graphs

A defect-d matching in a graph G is a matching which covers all but d vertices of G. G is d-covered if each edge of G belongs to a defect-d matching. Here we characterise d-covered graphs and d-covered connected bipartite graphs. We show that a regular graph G of degree r which is (r ? 1)-edge-connected is 0-covered or 1-covered depending on whether G has an even or odd number of vertices, but, given any non-negative integers r and d, there exists a graph regular of degree r with connectivity and edge-connectivity r ? 2 which does not even have a defect-d matching. Finally, we prove that a vertex-transitive graph is 0-covered or 1-covered depending on whether it has an even or odd number of vertices.     

Spanning Star Trees in Regular Graphs

For a subset W of vertices of an undirected graph G, let S(W) be the subgraph consisting of W, all edges incident to at least one vertex in W, and all vertices adjacent to at least one vertex in W. If S(W) is a tree containing all the vertices of G, then we call it a spanning star tree of G. In this case W forms a weakly connected but strongly acyclic dominating set for G. We prove that for every r ≥ 3, there exist r-regular n-vertex graphs that have spanning star trees, and there exist r-regular n-vertex graphs that do not have spanning star trees, for all n sufficiently large (in terms of r). Furthermore, the problem of determining whether a given regular graph has a spanning star tree is NP-complete.     

Path-path Ramsey-type numbers for the complete bipartite graph

For a fixed pair of integers r, s ≥ 2, all positive integers m and n are determined which have the property that if the edges of K_m,n (a complete bipartite graph with parts n and m) are colored with two colors, then there will always exist a path with r vertices in the first color or a path with s vertices in the second color.     

Closest 4-leaf power is fixed-parameter tractable

The NP-complete Closest 4-Leaf Power problem asks, given an undirected graph, whether it can be modified by at most r edge insertions or deletions such that it becomes a 4-leaf power. Herein, a 4-leaf power is a graph that can be constructed by considering an unrooted tree—the 4-leaf root—with leaves one-to-one labeled by the graph vertices, where we connect two graph vertices by an edge iff their corresponding leaves are at distance at most 4 in the tree. Complementing previous work on Closest 2-Leaf Power and Closest 3-Leaf Power, we give the first algorithmic result for Closest 4-Leaf Power, showing that Closest 4-Leaf Power is fixed-parameter tractable with respect to the parameter r.     

Hypergraphs and abstract polygons

The notion of a graph has recently been generalized to include structures called hypergraphs which have two or more vertices per edge. A hypergraph is called 2-settled if each pair of distinct vertices is contained in at most one edge. A connected 2-settled hypergraph which has at least two edges through each vertex might be called an abstract polygon. Lemma: Every abstract polygon contains a cycle. Shephard and Coxeter have examined certain abstract polygons called regular complex polygons, each of which is denoted by a symbol p {q} r where there are p vertices on each edge and r edges through each vertex. Theorem: The girth of the non-starry regular complex polygon p {q} r is q. Thus, the number q is finally given a simple combinatoric interpretation.     

The Four-in-a-Tree Problem in Triangle-Free Graphs

The three-in-a-tree algorithm of Chudnovsky and Seymour decides in time O(n ⁴) whether three given vertices of a graph belong to an induced tree. Here, we study four-in- a-tree for triangle-free graphs. We give a structural answer to the following question: what does a triangle-free graph look like if no induced tree covers four given vertices? Our main result says that any such graph must have the “same structure”, in a sense to be defined precisely, as a square or a cube. We provide an O(nm)-time algorithm that given a triangle-free graph G together with four vertices outputs either an induced tree that contains them or a partition of V(G) certifying that no such tree exists. We prove that the problem of deciding whether there exists a tree T covering the four vertices such that at most one vertex of T has degree at least 3 is NP-complete.     

Bold graph drawings

When a graph is drawn in a classical manner, its vertices are shown as small disks and its edges with a positive width; zero-width edges and zero-size vertices exist only in theory. Let r denote the radius of the disks that show vertices and w the width of edges. We give a list of conditions that make such a drawing good and that apply to not necessarily planar graphs. We show that if r<w, a vertex must have constant degree for a drawing to satisfy the conditions, and if r?w, a vertex can have any degree. We also give an algorithm that, for a given drawing and values for r and w, determines whether the bold drawing satisfies the conditions. Furthermore, we show how to maximize r and/or w without violating the conditions in polynomial time.     

Paths with a given number of vertices from each partite set in regular multipartite tournaments

A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph.For c?2 we prove that a regular c-partite tournament with r?2 vertices in each partite set contains a directed path with exactly two vertices from each partite set. Furthermore, if c?4, then we will show that almost all regular c-partite tournaments D contain a directed path with exactly r-s vertices from each partite set for each given integer s∈N, if r is the cardinality of each partite set of D. Some related results are also presented.     

Hereditary dominating pair graphs

An asteroidal triple (AT) is a set of vertices such that each pair of vertices is joined by a path that avoids the neighborhood of the third. Every AT-free graph contains a dominating pair, a pair of vertices such that for every path between them, every vertex of the graph is within distance one of the path. We say that a graph is a hereditary dominating pair (HDP) graph if each of its connected induced subgraphs contains a dominating pair. In this paper we introduce the notion of frame HDP graphs in order to capture the structure of HDP graphs that contain asteroidal triples. We also determine the maximum diameter of frame HDP graphs.     

A bound for the number of vertices of a polytope with applications

We prove that the number of vertices of a polytope of a particular kind is exponentially large in the dimension of the polytope. As a corollary, we prove that an n-dimensional centrally symmetric polytope with O(n) facets has {ie1-1} vertices and that the number of r-factors in a k-regular graph is exponentially large in the number of vertices of the graph provided k≥2r+1 and every cut in the graph with at least two vertices on each side has more than k/r edges.     

Intersecting all edges of convex polytopes by planes

A characterization theorem is given for 3-dimensional convex polytopes Q having the following property: There exists a polytope P, isomorphic to Q, all edges of which can be cut by a pair of planes that miss all its vertices. The result yields an affirmative solution of a conjecture of B. Grünbaum.     

On the maximal order of cyclicity of antisymmetric directed graphs

A directed graph G without loops or multiple edges is said to be antisymmetric if for each pair of distinct vertices of G (say u and v), G contains at most one of the two possible directed edges with end-vertices u and v. In this paper we study edge-sets M of an antisymmetric graph G with the following extremal property: By deleting all edges of M from G we obtain an acyclic graph, but by deleting from G all edges of M except one arbitrary edge, we always obtain a graph containing a cycle. It is proved (in Theorem 1) that if M has the above mentioned property, then the replacing of each edge of M in G by an edge with the opposite direction has the same effect as deletion: the graph obtained is acyclic. Further we study the order of cyclicity of G (= theminimalnumberofedgesinsuchasetM) and the maximal order of cyclicity in an antisymmetric graph with given number n of vertices. It is shown that for n < 10 this number is equal to the maximal number of edge-disjoint circuits in the complete (undirected) graph with n vertices and for n = 10 (and for an infinite set of n's) the first number is greater than the latter.     

An Ore-type condition for arbitrarily vertex decomposable graphs

Let G be a graph of order n and r, 1≤r≤n, a fixed integer. G is said to be r-vertex decomposable if for each sequence (n₁,…,n_r) of positive integers such that n₁+?+n_r=n there exists a partition (V₁,…,V_r) of the vertex set of G such that for each i∈{1,…,r}, V_i induces a connected subgraph of G on n_i vertices. G is called arbitrarily vertex decomposable if it is r-vertex decomposable for each r∈{1,…,n}.In this paper we show that if G is a connected graph on n vertices with the independence number at most ⌈n/2⌉ and such that the degree sum of any pair of non-adjacent vertices is at least n−3, then G is arbitrarily vertex decomposable or isomorphic to one of two exceptional graphs. We also exhibit the integers r for which the graphs verifying the above degree-sum condition are not r-vertex decomposable.     

Adjacency on the order polytope with applications to the theory of fuzzy measures

In this paper we study the adjacency structure of the order polytope of a poset. For a given poset, we determine whether two vertices in the corresponding order polytope are adjacent. This is done through filters in the original poset. We also prove that checking adjacency between two vertices can be done in quadratic time on the number of elements of the poset. As particular cases of order polytopes, we recover the adjacency structure of the set of fuzzy measures and obtain it for the set of p-symmetric measures for a given indifference partition; moreover, we show that the set of p-symmetric measures can be seen as the order polytope of a quotient set of the poset leading to fuzzy measures. From this property, we obtain the diameter of the set of p-symmetric measures. Finally, considering the set of p-symmetric measures as the order polytope of a direct product of chains, we obtain some other properties of these measures, as bounds on the volume and the number of vertices on certain cases.     

Using Lagrangians of hypergraphs to find non-jumping numbers(II)

Let r?2 be an integer. A real number α∈[0,1) is a jump for r if for any ε>0 and any integer m?r, any r-uniform graph with n>n₀(ε,m) vertices and density at least α+ε contains a subgraph with m vertices and density at least α+c, where c=c(α)>0 does not depend on ε and m. A result of Erd?s, Stone and Simonovits implies that every α∈[0,1) is a jump for r=2. Erd?s asked whether the same is true for r?3. Frankl and Rödl gave a negative answer by showing an infinite sequence of non-jumping numbers for every r?3. However, there are a lot of unknowns on determining whether or not a number is a jump for r?3. In this paper, we find two infinite sequences of non-jumping numbers for r=4, and extend one of the results to every r?4. Our approach is still based on the approach developed by Frankl and Rödl.     

Total domination dot-stable graphs

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. Two vertices of G are said to be dotted (identified) if they are combined to form one vertex whose open neighborhood is the union of their neighborhoods minus themselves. We note that dotting any pair of vertices cannot increase the total domination number. Further we show it can decrease the total domination number by at most 2. A graph is total domination dot-stable if dotting any pair of adjacent vertices leaves the total domination number unchanged. We characterize the total domination dot-stable graphs and give a sharp upper bound on their total domination number. We also characterize the graphs attaining this bound.     

Two theorems on graceful trees

Let T be a tree on n vertices which are labelled by the integers in N = {1,2,…,n} such that each vertex of T is associated with a distinct number in N. The weight of an edge is defined to be the absolute value of the difference between the two numbers labelled at its end vertices. If the weights of all edges of T are distinct, we call T a graceful tree. In this note, two methods for constructing bigger graceful trees from a given one and a given pair of graceful trees are provided.     

On types of growth for graph-different permutations

We consider an infinite graph G whose vertex set is the set of natural numbers and adjacency depends solely on the difference between vertices. We study the largest cardinality of a set of permutations of [n] any pair of which differ somewhere in a pair of adjacent vertices of G and determine it completely in an interesting special case. We give estimates for other cases and compare the results in case of complementary graphs. We also explore the close relationship between our problem and the concept of Shannon capacity “within a given type.”     

The face pair of planar graphs

Harary and Kovacs [Smallest graphs with given girth pair, Caribbean J. Math. 1 (1982) 24-26] have introduced a generalization of the standard cage question—r-regular graphs with given odd and even girth pair. The pair (ω,ε) is the girth pair of graph G if the shortest odd and even cycles of G have lengths ω and ε, respectively, and denote the number of vertices in the (r,ω,ε)-cage by f(r,ω,ε). Campbell [On the face pair of cubic planar graph, Utilitas Math. 48 (1995) 145-153] looks only at planar graphs and considers odd and even faces rather than odd and even cycles. He has shown that f(3,ω,4)=2ω and the bounds for the left cases. In this paper, we show the values of f(r,ω,ε) for the left cases where (r,ω,ε)∈{(3,3,ε),(4,3,ε),(5,3,ε), (3,5,ε)}.