Regular subgraphs of almost regular graphs

Suppose every vertex of a graph G has degree k or k + 1 and at least one vertex has degree k + 1. It is shown that if k ≥ 2q ? 2 and q is a prime power then G contains a q-regular subgraph (and hence an r-regular subgraph for all r < q, r ≡ q (mod 2)). It is also proved that every simple graph with maximal degree Δ ≥ 2q ? 2 and average degree $\text{d > (}\text{(2q ? 2)}\text{(2q ? 1)}\text{)(Δ + 1)}$, where q is a prime power, contains a q-regular subgraph (and hence an r-regular subgraph for all r < q, r ≡ q (mod 2)). These results follow from Chevalley's and Olson's theorems on congruences.     

Packing vertices and edges in random regular graphs

In this paper we consider the problem of finding large collections of vertices and edges satisfying particular separation properties in random regular graphs of degree r, for each fixed r ≥ 3. We prove both constructive lower bounds and combinatorial upper bounds on the maximal sizes of these sets. The lower bounds are proved by analyzing a class of algorithms that return feasible solutions for the given problems. The analysis uses the differential equation method proposed by Wormald [Lectures on Approximation and Randomized Algorithms, PWN, Wassaw, 1999, pp. 239–298]. The upper bounds are proved by direct combinatorial means. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Efficient edge domination in regular graphs

An induced matching of a graph G is a matching having no two edges joined by an edge. An efficient edge dominating set of G is an induced matching M such that every other edge of G is adjacent to some edge in M. We relate maximum induced matchings and efficient edge dominating sets, showing that efficient edge dominating sets are maximum induced matchings, and that maximum induced matchings on regular graphs with efficient edge dominating sets are efficient edge dominating sets. A necessary condition for the existence of efficient edge dominating sets in terms of spectra of graphs is established. We also prove that, for arbitrary fixed p≥3, deciding on the existence of efficient edge dominating sets on p-regular graphs is NP-complete.     

Delaunay graphs are almost as good as complete graphs

LetS be any set ofN points in the plane and let DT(S) be the graph of the Delaunay triangulation ofS. For all pointsa andb ofS, letd(a, b) be the Euclidean distance froma tob and let DT(a, b) be the length of the shortest path in DT(S) froma tob. We show that there is a constantc (≤((1+√5)/2) π≈5.08) independent ofS andN such that $$\frac{{DT(a,b)}}{{d(a,b)}}< c.$$      

Amalgamations of almost regular edge-colourings of simple graphs

A finite graph F is a detachment of a finite graph G if G can be obtained from F by partitioning V(F) into disjoint sets S₁, …, S_n and identifying the vertices in S_i to form a single vertex α_i for i = 1, …, n. Thus E(F) = E(G) and an edge which joins an element of S_i to an element of S_j in F will join α_i to α_j in G. If L is a subset of E(G) then G(L) denotes the subgraph of G such that V(G(L)) = V(G), E(G(L)) = L. We call a graph almost regular if there is an integer d such that every vertex has valency d or d + 1. Suppose that E(G) is partitioned into disjoint sets E₁, …, E_r. Hilton [3] found necessary and sufficient conditions for the existence of a detachment F of G such that F is a complete graph with 2r + 1 vertices and F(E_i) is a Hamilton circuit of F for i = 1, …, r. We give a new proof of Hilton's theorem, which also yields a generalisation. Specifically, for any q ∈ {0, 1, …, r}, we find necessary and sufficient conditions for G to have a detachment F without loops or multiple edges such that F(E₁), …, F(E_r) are almost regular and F(E₁), …, F(E_q) are 2-edge-connected and each vertex ξ of G arises by identification from a prescribed number g(ξ) of vertices of F.     

Geodesics and almost geodesic cycles in random regular graphs

A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance d_G(u, v) is at least d_C(u, v)?e(n). Let ω(n) be any function tending to infinity with n. We consider a random d‐regular graph on n vertices. We show that almost all pairs of vertices belong to an almost geodesic cycle C with e(n) = log_d?1log_d?1n+ ω(n) and |C| = 2log_d?1n+ O(ω(n)). Along the way, we obtain results on near‐geodesic paths. We also give the limiting distribution of the number of geodesics between two random vertices in this random graph. Copyright © 2010 John Wiley & Sons, Ltd. J Graph Theory 66:115‐136, 2011     

Almost regular edge colorings and regular decompositions of complete graphs

For k = 1 and k = 2, we prove that the obvious necessary numerical conditions for packing t pairwise edge‐disjoint k‐regular subgraphs of specified orders m₁,m₂,… ,m_t in the complete graph of order n are also sufficient. To do so, we present an edge‐coloring technique which also yields new proofs of various known results on graph factorizations. For example, a new construction for Hamilton cycle decompositions of complete graphs is given. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 499–506, 2008     

On almost regular automorphisms

Among other things, we prove that a polycyclic group admitting an automorphism of order two with finitely many fixed elements is abelian-by-finite. An example shows that this result cannot be extended to finitely generated soluble groups.     

On almost self-complementary graphs

A graph is called almost self-complementary if it is isomorphic to one of its almost complements X^c-I, where X^c denotes the complement of X and I a perfect matching (1-factor) in X^c. Almost self-complementary circulant graphs were first studied by Dobson and Šajna [Almost self-complementary circulant graphs, Discrete Math. 278 (2004) 23-44]. In this paper we investigate some of the properties and constructions of general almost self-complementary graphs. In particular, we give necessary and sufficient conditions on the order of an almost self-complementary regular graph, and construct infinite families of almost self-complementary regular graphs, almost self-complementary vertex-transitive graphs, and non-cyclically almost self-complementary circulant graphs.     

On almost distance-regular graphs

Distance-regular graphs are a key concept in Algebraic Combinatorics and have given rise to several generalizations, such as association schemes. Motivated by spectral and other algebraic characterizations of distance-regular graphs, we study ‘almost distance-regular graphs’. We use this name informally for graphs that share some regularity properties that are related to distance in the graph. For example, a known characterization of a distance-regular graph is the invariance of the number of walks of given length between vertices at a given distance, while a graph is called walk-regular if the number of closed walks of given length rooted at any given vertex is a constant. One of the concepts studied here is a generalization of both distance-regularity and walk-regularity called m-walk-regularity. Another studied concept is that of m-partial distance-regularity or, informally, distance-regularity up to distance m. Using eigenvalues of graphs and the predistance polynomials, we discuss and relate these and other concepts of almost distance-regularity, such as their common generalization of (?,m)-walk-regularity. We introduce the concepts of punctual distance-regularity and punctual walk-regularity as a fundament upon which almost distance-regular graphs are built. We provide examples that are mostly taken from the Foster census, a collection of symmetric cubic graphs. Two problems are posed that are related to the question of when almost distance-regular becomes whole distance-regular. We also give several characterizations of punctually distance-regular graphs that are generalizations of the spectral excess theorem.     

Occupation games on graphs in which the second player takes almost all vertices

Given a connected graph G=(V,E), two players take turns occupying vertices v∈V by placing black and white tokens so that the current vertex sets B,W⊆V are disjoint, B∩W=0?, and the corresponding induced subgraphs G[B] and G[W] are connected any time. A player must pass whenever (s)he has no legal move. (Obviously, after this, the opponent will take all remaining vertices, since G is assumed connected.) The game is over when all vertices are taken, V=B^∗∪W^∗. Then, Black and White get b=|B^∗|/|V| and w=|W^∗|/|V|, respectively. Thus, the occupation game is one-sum, b+w=1, and we could easily reduce it to a zero-sum game by simply shifting the payoffs, b^′=b−1/2,w^′=w−1/2. Let us also notice that b≥0 and w≥0; moreover, b>0 and w>0 whenever |V|>1.[Let us remark that the so-called Chinese rules define similar payoffs for the classic game of GO, yet, the legal moves are defined in GO differently.]Like in GO, we assume that Black begins. It is easy to construct graphs in which Black can take almost all vertices, more precisely, for each ε>0 there is a graph G for which b>1−ε. In this paper we show that, somewhat surprisingly, there are also graphs in which White can take almost all vertices.     

On Hamiltonian-connected regular graphs

In this paper it is shown that any m-regular graph of order 2m (m ≧ 3), not isomorphic to K_m,m, or of order 2m + 1 (m even, m ≧ 4), is Hamiltonian connected, which extends a previous result of Nash-Williams. As a corollary, it is derived that any such graph contains atleast m Hamiltonian cycles for odd m and atleast 1/2m Hamiltonian cycles for even m.     

On quasi-strongly regular graphs

We study the quasi-strongly regular graphs, which are a combinatorial generalization of the strongly regular and the distance regular graphs. Our main focus is on quasi-strongly regular graphs of grade 2. We prove a “spectral gap”-type result for them which generalizes Seidel's well-known formula for the eigenvalues of a strongly regular graph. We also obtain a number of necessary conditions for the feasibility of parameter sets and some structural results. We propose the heuristic principle that the quasi-strongly regular graphs can be viewed as a “lower-order approximation” to the distance regular graphs. This idea is illustrated by extending a known result from the distance-regular case to the quasi-strongly regular case. Along these lines, we propose a number of conjectures and open problems. Finally, we list the all the proper connected quasi-strongly graphs of grade 2 with up to 12 vertices.     

On quasi-strongly regular graphs

We study the quasi-strongly regular graphs, which are a combinatorial generalization of the strongly regular and the distance regular graphs. Our main focus is on quasi-strongly regular graphs of grade 2. We prove a “spectral gap”-type result for them which generalizes Seidel's well-known formula for the eigenvalues of a strongly regular graph. We also obtain a number of necessary conditions for the feasibility of parameter sets and some structural results. We propose the heuristic principle that the quasi-strongly regular graphs can be viewed as a “lower-order approximation” to the distance regular graphs. This idea is illustrated by extending a known result from the distance-regular case to the quasi-strongly regular case. Along these lines, we propose a number of conjectures and open problems. Finally, we list the all the proper connected quasi-strongly graphs of grade 2 with up to 12 vertices.