Hamilton Cycle

Combinatorial Chemistry is a powerful new technology in drug design and molecular recognition. It is a wetlaboratory methodology aimed at "massively parallel" screening of chemical compounds for the dkcovery of compounds that have a certain biological activity. The power of the method comes from the interaction between experimental design and computational modeling. Principles of "rational" drug design are used in the construction of combinatorial libraries to speed up the discovery of lead compounds with the desired biological activity. This paper presents algorithms, software development and computational complexity analysis for problems arising in the design of combinatorial libraries for drug discovery. We provide exact polynomial time algorithms and intractability results for several Inverse Problems -formulated as (chemical) graph reconstruction problems -related to the design of combinatorial libraries. These are the first rigorous algorithmic results in the literature. We also present results provided by our combinatorial chemistry software package OCOTILLO for combinatorial peptide design using real data libraries. The package provides exact solutions for general inverse problems based on shortest-path topological indices. Our results are superior both in accuracy and computing time to the best software reports published in the literature. For 5-peptoid design, the computation is rigorously reduced to an exahustive search of about 2% of the search space; the exact solutions are found in a few minutes.

Let A be an m × n matrix in which the entries of each row are all distinct. Drisko [4] showed that, if m ≥ 2n -1, then A has a transversal : a set of n distinct entries with no two in the same row or column. We generalize this to matrices with entries in a matroid. For such a matrix A, we show that if each row of A forms an independent set, then we can require the transversal to be independent as well. We determine the complexity of an algorithm based on the proof of this result. Lastly, we observe that m ≥ 2n -1 appears to force the existence of not merely one but many transversals. We discuss a number of conjectures related to this observation (some of which involve matroids and some of which do not).

We address the problem of minimizing the communication involved in the exchange of similar documents. We consider two users, A and B, who hold documents x and y respectively. Neither of the users has any information about the other's document. They exchange messages so that B computes x; it may be required that A compute y as well. Our goal is to design communication protocols with the main objective of minimizing the total number of bits they exchange; other objectives are minimizing the number of rounds and the complexity of internal computations. An important notion which determines the efficiency of the protocols is how one measures the distance between x and y. We consider several metrics for measuring this distance, namely the Hamming metric, the Levenshtein metric (edit distance), and a new LZ metric, which is introduced in this paper. We show how to estimate the distance between x and y using a single message of logarithmic size. For each metric, we present the first communication-efficient protocols, which often match the corresponding lower bounds. In consequence, we obtain error-correcting codes for these error models which correct up to d errors in n characters using O(d•polylog(n)) bits. Our most interesting methods use a new transformation from LZ distance to Hamming distance.

A (κ, τ )-regular set is a vertex subset S inducing a κ-regular subgraph such that every vertex out of S has τ neighbors in S. This article is an expository overview of the main results obtained for graphs with (κ, τ )-regular sets. The graphs with classical combinatorial structures, like perfect matchings, Hamilton cycles, efficient dominating sets, etc, are characterized by (κ, τ )-sets whose determination is equivalent to the determination of those classical combinatorial structures. The characterization of graphs with these combinatorial structures are presented. The determination of (κ, τ )-regular sets in a finite number of steps is deduced and the main spectral properties of these sets are described.

Let G be a graph of order n and let u, v be vertices of G. Let κG(u, v) denote the maximum number of internally disjoint u–v paths in G. Then the average connectivity κ(G) of G, is defined as κ(G) = ∑ {u,v}⊆V (G) κG(u, v)/ ( n 2 ) . If k ≥ 1 is an integer, then G is minimally kconnected if κ(G) = k and κ(G− e) < k for every edge e of G. We say that G is an optimal minimally k-connected graph if G has maximum average connectivity among all minimally k-connected graphs of order n. Based on a recent structure result for minimally 2-connected graphs we conjecture that, for every integer k ≥ 3, if G is an optimal minimally k-connected graph of order n ≥ 2k + 1, then G is bipartite, with the set of vertices of degree k and the set of vertices of degree exceeding k as its partite sets. We show that if this conjecture is true, then κ(G) < 98k for every minimally k-connected graph G. For every k ≥ 3, we describe an infinite family of minimally k-connected graphs whose average connectiv...

When ${\bf t} := \langle t_1,t_2,\ldots,t_k\rangle$ is a sequence of transpositions on the finite set $n:=\{0,1,\ldots,n-1\}$, then $\bigcirc{\bf t}:= t_1\circ t_2\circ\cdots\circ t_k$ denotes the compositional product of the sequence. Our paper treats the set ${\rm Prod}({\bf t})$ of all $\bigcirc{\bf s}$, where ${\bf s}$ is a sequence obtained by rearranging the terms of ${\bf t}$. The paper characterizes the set of all transpositional sequences ${\bf t}$ for which ${\rm Prod}({\bf t})$ is the subset of a single congugacy class in the symmetric group ${\rm Sym}(n)$; we call such ${\bf t}$ {\it conjugacy invariant}. At the opposite extreme, the paper studies conditions under which ${\bf t}$ is {\it permutationally complete}, which is to say, those ${\bf t}$ for which either ${\rm Prod}({\bf t}) = {\rm Alt}(n)$ or ${\rm Prod}({\bf t}) = {\rm Sym}(n)\setminus{\rm Alt}(n)$.

It is shown that every connected, locally connected graph with the maximum vertex degree Δ(G) = 5 and the minimum vertex degree δ(G) ≥ 3 is fully cycle extendable. For Δ(G) ≤ 4, all connected, locally connected graphs, including infinite ones, are explicitly described. The Hamilton Cycle problem for locally connected graphs with Δ(G) ≤ 7 is shown to be NP-complete.

A triangulated d-manifold K, satisfies the inequality f0(K)−d−1 2 ≥ d+2 2 β 1 (K; Z 2) for d ≥ 3. The triangulated d-manifolds that meet the bound with equality are called tight neighborly. In this paper, we present tight neighborly triangulations of 4-manifolds on 15 vertices with Z 3 as automorphism group. One such example was constructed by Bagchi and Datta in 2011. We show that there are exactly 12 such triangulations up to isomorphism, 10 of which are orientable.

We complete the proof of the Friedlander, Gordon and Miller Conjecture that every finite abelian group whose Sylow 2-subgroup either is trivial or both non-trivial and non-cyclic is R-sequenceable. This settles a question of Ringel for abelian groups.

In this paper, we prove that the wrapped Butterfly digraph ?T&F(d,n) of degree d and dimension n contains at least d-1 arc-disjoint Hamilton circuits, answering a conjecture of Barth [5]. We also conjecture that W&g(d,n) can be decomposed into d Hamilton circuits, except for {d = 2 and n = 2}, {d = 2 and n = 3) and {d = 3 and n = 2). We show that it suffices to prove this conjecture for d prime and n = 2. Then, we give such a Hamilton decomposition for all primes to 12 000 by a clever computer search, and so, as a corollary, we have a Hamilton decomposition of W&F(d, n) for any d divisible by a number q, with 4 <q < 12 000.

Cycle pre x digraphs are a class of Cayley coset graphs with many remarkable properties such as symmetry, large number of nodes for a given degree and diameter, simple shortest path routing, Hamiltonicity, optimal connectivity, and others. In this paper we show that the cycle pre x digraphs, like the Kautz digraphs, contain cycles of all lengths l, with l between two and N, the order of the digraph, except for N ; 1.

A classic result of G. A. Dirac in graph theory asserts that every n-vertex graph (n ≥ 3) with minimum degree at least n/2 contains a spanning (so-called Hamilton) cycle. G. Y. Katona and H. A. Kierstead suggested a possible extension of this result for k-uniform hypergraphs. There a Hamilton cycle of an n-vertex hypergraph corresponds to an ordering of the vertices such that every k consecutive (modulo n) vertices in the ordering form an edge. Moreover, the minimum degree is the minimum (k − 1)-degree, i.e. the minimum number of edges containing a fixed set of k − 1 vertices. V. Rödl, A. Ruciński, and E. Szemerédi verified (approximately) the conjecture of Katona and Kierstead and showed that every n-vertex, k-uniform hypergraph with minimum (k − 1)-degree (1/2 + o(1))n contains such a tight Hamilton cycle. We study the similar question for Hamilton-cycles. A Hamilton-cycle in an n-vertex, k-uniform hypergraph (1 ≤ < k) is an ordering of the vertices and an ordered subset of the edges such that each such edge corresponds to k consecutive (modulo n) vertices and two consecutive edges intersect in precisely vertices. We prove sufficient minimum (k − 1)-degree conditions for Hamiltoncycles if < k/2. In particular, we show that for every < k/2 every n-vertex, k-uniform hypergraph with minimum (k − 1)-degree (1/(2(k −)) + o(1))n contains such a loose Hamilton-cycle. This degree condition is approximately tight and was conjectured by D. Kühn and D. Osthus (for = 1), who verified it when k = 3. Our proof is based on the so-called weak regularity lemma for hypergraphs and follows the approach of V. Rödl, A. Ruciński, and E. Szemerédi.

The middle-levels graph $M_k$ ($0<k\in\mathbb{Z}$) has a dihedral quotient graph $R_k$ whose vertices represent all $k$-edge ordered (rooted) trees $T$. Each $T$ is encoded as a $(2k+1)$-string $F(T)$ that annotates by means of DFS the descending nodes via Kierstead-Trotter lexical colors $0,\ldots,k$, and the ascending edges via $k$ asterisks. This $T$ is assigned a restricted-growth $k$-string in two ways. One of these two ways is obtained via nested substring-swaps in the $F(T)$'s (i.e. pruning-and-regrafting in the $T$'s). These swaps allow sorting the vertices of all $R_k$'s in a canonical, natural, enumerative way that unifies the presentation of the $M_k$'s and their Hamilton cycles by T. Mütze et al.

A system of numeration in which every $k$, with $0<k\in\mathbb{Z}$, appears as a restricted growth string, or RGS, has the $k$-th Catalan number as the RGS $10^k$. This induces a canonical ordering of the vertices of the dihedral quotients of the middle-levels graphs.

A graph G = (V , E) is called a split graph if there exists a partition V = I ∪ K such that the subgraphs of G induced by I and K are empty and complete graphs, respectively. In 1980, Burkard and Hammer gave a necessary but not sufficient condition for hamiltonian split graphs with |I | < |K|. In this paper, we show that the Burkard-Hammer condition is also sufficient for the existence of a Hamilton cycle in a split graph G such that 5 = |I | < |K| and the minimum degree (G) |I | − 3. For the case 5 = |I | < |K|, all split graphs satisfying the Burkard-Hammer condition but having no Hamilton cycles are also described.

Stormwater runoff in Washington, D.C. frequently exceeds the capacity of the combined sewer system. This overflow leads to periodic flooding of the National Mall and low-lying areas, and contributes to pollution of the region's rivers and streams. By capturing, collecting and reusing stormwater on site, runoff is prevented from entering the city's system, helping to reduce flooding and the problems it causes. None of the streetscaped areas are irrigated. Captured stormwater supplies all of the courtyard irrigation and replenishes the water feature. Stormwater is collected from the 40,000 ft 2 area of the office-building roof and the courtyard and directed into the filtration system and 7,500 gallon storage cistern located in the underground parking garage. Collected and filtered water is then pumped back to the courtyard for irrigation and to replace water lost to evaporation in the water feature. To determine the total amount of water reused annually, and therefore potable water saved, the amount of water needed per year for irrigation was added to the amount of water lost per year due to evaporation.

Stormwater runoff in Washington, D.C. frequently exceeds the capacity of the combined sewer system. This overflow leads to periodic flooding of the National Mall and low-lying areas, and contributes to pollution of the region's rivers and streams. By capturing, collecting and reusing stormwater on site, runoff is prevented from entering the city's system, helping to reduce flooding and the problems it causes. None of the streetscaped areas are irrigated. Captured stormwater supplies all of the courtyard irrigation and replenishes the water feature. Stormwater is collected from the 40,000 ft 2 area of the office-building roof and the courtyard and directed into the filtration system and 7,500 gallon storage cistern located in the underground parking garage. Collected and filtered water is then pumped back to the courtyard for irrigation and to replace water lost to evaporation in the water feature. To determine the total amount of water reused annually, and therefore potable water saved, the amount of water needed per year for irrigation was added to the amount of water lost per year due to evaporation.

It is shown that a minimal normal subgroup of a transitive permutation group of square-free degree in its induced action is simple and quasiprimitive, with three exceptions related to A 5 , A 7 , and PSL(2, 29). Moreover, it is shown that a minimal normal subgroup of a 2-closed permutation group of square-free degree in its induced action is simple. As an almost immediate consequence, it follows that a 2-closed transitive permutation group of square-free degree contains a semiregular element of prime order, thus giving a partial affirmative answer to the conjecture that all 2-closed transitive permutation groups contain such an element (see [D. Marušič, On vertex symmetric digraphs,

In this paper we determine the positive integers n and k for which there exists a homogeneous factorisation of a complete digraph on n vertices with k 'common circulant' factors. This means a partition of the arc set of the complete digraph K n into k circulant factor digraphs, such that a cyclic group of order n acts regularly on the vertices of each factor digraph whilst preserving the edges, and in addition, an overgroup of this permutes the factor digraphs transitively amongst themselves. This determination generalises a previous result for self-complementary circulants.

Let P be a graph property. A graph G is said to be locally P if the subgraph induced by the open neighbourhood of every vertex in G has property P. Ryjáček's well-known conjecture that every connected, locally connected graph is weakly pancyclic motivated us to consider the global cycle structure of locally P graphs, where P is the property of having diameter at most k for some fixed k ≥ 1. For k = 2 these graphs are called locally isometric graphs. For ∆ ≤ 5, we obtain a complete structural characterization of locally isometric graphs that are not fully cycle extendable. For ∆ = 6, it is shown that locally isometric graphs that are not fully cycle extendable contain a pair of true twins of degree 6. Infinite classes of locally isometric graphs with ∆ = 6 that are not fully cycle extendable are described and observations are made that suggest that a complete characterization of these graphs is unlikely. It is shown that Ryjáček's conjecture holds for all locally isometric graphs with ∆ ≤ 6. The Hamilton cycle problem for locally isometric graphs with maximum degree at most 8 is shown to be NP-complete.

In 1987, Akers, Harel and Krishnamurthy proposed the star graph Σ(n) as a new topology for interconnection networks. Hamiltonian properties of these graphs have been investigated by several authors. In this paper, we prove that Σ(n) contains n/8 pairwise edge-disjoint Hamilton cycles when n is prime, and Ω(n/ log log n) such cycles for arbitrary n.

We deal with Oberwolfach factorizations of the complete graphs K n and K n *, which admit a regular group of automorphisms. We show that the existence of such a factorization is equivalent to the existence of a certain difference sequence defined on the elements of the automorphism group, or to a certain sequencing of the elements of that group. In the particular case of a hamiltonian factorization of the directed graph K n * which admits a regular group of automorphisms G (|G|=n&1), we have that such a factorization exists if and only if G is sequenceable. We shall demonstrate how the mentioned above (difference) sequences may be used in the construction of such factorizations. We prove also that a hamiltonian factorization of the undirected graph K n (n odd) which admits a regular group of automorphisms G (|G| =(n&1)Â2) exists if and only if n#3 (mod 4), without further restrictions on the structure of G.

Related to Chvfital's famous conjecture stating that every 2-tough graph is hamiltonian, we study the relation of toughness and hamiltonieity on special classes of graphs. First, we consider properties of graph classes related to hamiltonicity, traceability and toughness concepts and display some algorithmic consequences. Furthermore, we present a polynomial time algorithm deciding whether the toughness of a given split graph is less than one and show that deciding whether the toughness of a bipartite graph is exactly one is coNP-complete. We show that every 3-tough split graph is hamiltonian and that there is a sequence of nonhamiltonian split graphs with toughness converging to 3.

Let p be an odd prime number. This work applies some group concepts to construct the Wreath Product of two permutation groups of prime degrees. We used numerical approach to investigate and determine the primitive and regular nature of the constructed Wreath Product Group of degree 5p. We apply Computational Group Theory (GAP) to facilitate as well as validate our results.

Let 0 < k ∈ Z. The anchored Dyck words of length n = 2k+1 (obtained by prefixing a 0-bit to each Dyck word of length 2k and used to reinterpret the Hamilton cycles in the odd graph O k and the middle-levels graph M k found by Mütze et al.) represent in O k (resp., M k) the cycles of an n-(resp., 2n-) 2-factor and its cyclic (resp., dihedral) vertex classes, and are equivalent to Dyck-nest signatures. A sequence is obtained by updating these signatures according to the depth-first order of a tree of restricted growth strings (RGS's), reducing the RGS-generation of Dyck words by collapsing to a single update the time-consuming i-nested castling used to reach each non-root Dyck word or Dyck nest. This update is universal, for it does not depend on k.

Let 0 < k ∈ Z. A reinterpretation of the proof of existence of Hamilton cycles in the middle-levels graph M k induced by the vertices of the (2k + 1)-cube representing the k-and (k + 1)-subsets of {0,. .. , 2k} is given via an associated dihedral quotient graph of M k whose vertices represent the ordered (rooted) trees of order k + 1 and size k.

Let G be a finite group, and let 1 G ∈ S ⊆ G. A Cayley di-graph Γ = Cay(G, S) of G relative to S is a di-graph with a vertex set G such that, for x, y ∈ G, the pair (x, y) is an arc if and only if yx −1 ∈ S. Further, if S = S −1 := {s −1 |s ∈ S}, then Γ is undirected. Γ is conected if and only if G = s. A Cayley (di)graph Γ = Cay(G, S) is called normal if the right regular representation of G is a normal subgroup of the automorphism group of Γ. A graph Γ is said to be arc-transitive, if Aut(Γ) is transitive on an arc set. Also, a graph Γ is s-regular if Aut(Γ) acts regularly on the set of s-arcs. In this paper, we first give a complete classification for arc-transitive Cayley graphs of valency five on finite Abelian groups. Moreover, we classify s-regular Cayley graph with valency five on an abelian group for each s ≥ 1.

In Pakistan, many subsurface (SS) drainage projects were launched by the Salinity Control and Reclamation Project (SCARP) to deal with twin problems (waterlogging and salinity). In some cases, sump pumps were installed for the disposal of SS effluent into surface drainage channels. Presently, sump pumps have become dysfunctional due to social and financial constraints. This study evaluates the alternate design of the Paharang drainage system that could permit the discharge of the SS drainage system in the response of gravity. The proposed design was completed after many successive trials in terms of lowering the bed level and decreasing the channel bed slope. Interconnected MS-Excel worksheets were developed to design the L-section and X-section. Design continuity of the drainage system was achieved by ensuring the bed and water levels of the receiving drain were lower than the outfalling drain. The drain cross-section was set within the present row with a few changes on the service...

A vertex of degree one is called an end-vertex and the set of end-vertices of G is denoted by End(G). For a positive integer k, a tree T be called k-ended tree if | End(T) |≤ k. In this paper, for each 3-regular connected graph with |G| 8, we find a positive integer k related to order of G, such that G has a spanning k-ended tree, and with construction a sequence of 3-regular connected graphs which their orders approach to infinity with known minimum possible number of end-vertices of their spanning trees. At the end, we given a conjecture about spanning k-ended trees of 3-regular connected graphs.

We show that the 56-vertex Klein cubic graph Γ ′ can be obtained from the 28-vertex Coxeter cubic graph Γ by 'zipping' adequately the squares of the 24 7-cycles of Γ endowed with an orientation obtained by considering Γ as a C-ultrahomogeneous digraph, where C is the collection formed by both the oriented 7-cycles C7 and the 2-arcs P3 that tightly fasten those C7 in Γ. In the process, it is seen that Γ ′ is a C ′-ultrahomogeneous (undirected) graph, where C ′ is the collection formed by both the 7-cycles C7 and the 1-paths P2 that tightly fasten those C7 in Γ ′. This yields an embedding of Γ ′ into a 3-torus T3 which forms the Klein map of Coxeter notation (7, 3)8. The dual graph of Γ ′ in T3 is the distance-regular Klein quartic graph, with corresponding dual map of Coxeter notation (3, 7)8.

Let 0 < k ∈ Z. A reinterpretation of the proof of existence of Hamilton cycles in the middle-levels graph M k induced by the vertices of the (2k + 1)-cube representing the k-and (k + 1)-subsets of {0,. .. , 2k} is given via an associated dihedral quotient graph of M k whose vertices represent the ordered (rooted) trees of order k + 1 and size k.

Let $0<k\in\mathbb{Z}$ and let $M_k$ be the subgraph of the Boolean lattice induced by the $k$- and $(k+1)$-levels of the Boolean lattice on $2k+1$ elements. The introduction of a numeral system based on the $k$-th Catalan numbers $C_k={2k+1\choose k}/(2k+1)$, that allows to linearly classify the vertices of a folding of a quotient graph of $M_k$, reveals two extremal approaches to Hável Conjecture on the existence of Hamilton cycles in the $M_k$s.

Let $m=2k+1\in\Z$ be odd. The $m$-cube graph $H_m$ is the Hasse diagram of the Boolean lattice on the coordinate set $\Z_m$. A rooted binary tree $T$ is introduced that has as its nodes the translation classes mod $m$ of the weight-$k$ vertices of all $H_m$, for $0<k\in\Z$, with an equivalent form of $T$ whose nodes are the translation classes mod $m$ of weight-$(k+1)$ vertices via complemented reversals of the former class representatives, both forms of $T$ expressible uniquely via the Kierstead-Trotter lexical 1-factorizations of the middle-levels graphs $M_k$. This yields a linear order for both middle levels of any $H_m$ via left-to-right concatenation of the right descending-paths of the size-$m$ nodes of $T$. The tree $T$, of interest on its own and whose structure is ruled by Catalan's triangle, has an inductive definition by means of the said 1-factorizations, which allows to transform each claimed linear order into an adjacency list expressible via a $\frac{1}{k}{m\choose k}\times(k+1)$-matrix whose columns are lexically colored in $\{0,...,k\}$, yielding a Hamilton-cycle codification for $M_k$.

A system of numeration in which every $k$, with $0<k\in\mathbb{Z}$, appears as a restricted growth string, or RGS, has the $k$-th Catalan number as the RGS $10^k$. This induces a linear ordering of the vertices of the dihedral quotients of the middle-levels graphs.

Some remarks on the middle-levels graphs, their theorem on the existence of Hamilton cycles by T. Mütze and the short proof of this by P. Gregor, T. Mütze and J. Nummenpalo are provided by means of restricted-growth strings. This allows to present a construction as in the cited proof by combining the cycles provided by the union of the 1-factors of the first two lexical colors, 0 and 1, and certain 6-cycles that connect them, an alternative visualizing methodology of that used originally.

As green roofs, terraces, and walls are becoming more common, structural engineers appear to be unaware of the structural issues involved and how to address them. Green roofs, terraces, and walls are an architectural/mechanical approach that tackles the sustainable design issues of storm water runoff, reduction of building energy use, and an opportunity to provide usable space to building occupants. Structural engineers must understand the structural implications of such approaches with regards to static loads, dynamic loads, serviceability, durability, and anchorage. This document describes the structural implications of intensive green roofs/terraces, extensive green roofs, and green walls. An in depth discussion on assumed dead loads, live loads, seismic loads, wind effects, load combinations, serviceability concerns, and ASTM standards is provided. An analysis of tree loading, sloped roofs, seismic anchorage of green roofs, and recommended structural design specifications and strategies will also be presented. Lastly, strategies utilizing green roofs within the context of the sustainable metric systems such as USBGC's LEED rating system will be addressed. This document will provide a resource for engineers looking to easily, safely, and effectively facilitate the integration of green roofs into their projects. Container Size Container Grown/Field Grown Weights-lbs (kg)

A graph G is collapsible if for every even subset X ⊆ V (G), G has a subgraph Γ such that G − E(Γ) is connected and the set of odd-degree vertices of Γ is X. A graph obtained by contracting all the non-trivial collapsible subgraphs of G is called the reduction of G. In this paper, we characterize graphs of diameter two in terms of collapsible subgraphs and investigate the relationship between the line graph of the reduction and the reduction of the line graph. Our results extend former results in [H.-J. Lai, Reduced graph of diameter two, J. Graph Theory 14 (1) (1990) 77-87], and in [P.A. Catlin, Iqblunnisa, T.N. Janakiraman, N. Srinivasan, Hamilton cycles and closed trails in iterated line graphs, J. Graph Theory 14 (1990) 347-364].

For an integer k > 0, a graph G is k-triangular if every edge of G lies in at least k distinct 3-cycles of G. In (J Graph Theory 11:399-407 (1987)), Broersma and Veldman proposed an open problem: for a given positive integer k, determine the value s for which the statement "Let G be a k-triangular graph. Then L(G), the line graph of G, is s-hamiltonian if and only L(G) is (s + 2)-connected" is valid. Broersma and Veldman proved in 1987 that the statement above holds for 0 ≤ s ≤ k and asked, specifically, if the statement holds when s = 2k. In this paper, we prove that the statement above holds for 0 ≤ s ≤ max{2k, 6k − 16}.

A graph G of order n is pancyclic if it contains cycles of all lengths from 3 to n. A graph is called cycle extendable if for every cycle C of less than n vertices there is another cycle C * containing all vertices of C plus a single new vertex. Clearly, every cycle extendable graph is pancyclic if it contains a triangle. Cycle extendability has been intensively studied for dense graphs while little is known for sparse graphs, even very special graphs. We show that all Hamiltonian interval graphs are cycle extendable. This supports a conjecture of Hendry that all Hamiltonian chordal graphs are cycle extendable.

For s ≥ 3 a graph is K 1,s-free, if it does not contain an induced subgraph isomorphic to K 1,s. For s = 3, such graphs are called claw-free graphs. Results on disjoint cycles in claw-free graphs satisfying certain minimum degree conditions will be be discussed, such as if G is claw-free of sufficiently large order n = 3k with δ(G) ≥ n/2, then G contains k disjoint triangles. Also, the extension of results on disjoint cycles in claw-free graphs satisfying certain minimum degree conditions to K 1,s-free graphs for s > 3 will be presented. These results will be used to prove the existence of minimum degree conditions that imply the existence of powers Hamiltonian cycle in generalized claw-free graphs.

Let G be a finite abelian group of order n. The barycentric Ramsey number BR(H,G) is the minimum positive integer r such that any coloring of the edges of the complete graph Kr by elements of G contains a subgraph H whose assigned edge color constitutes a barycentric sequence, i.e. there exists one edge whose color is the &quot;average&quot; of

Rain gardens are commonly used as green infrastructure to reduce runoff volume and introduce ecological benefits to urban areas. The objective of this study was to evaluate hydrologic performance of a rain garden constructed on low-permeability subsoil with three different drainage systems: no drain, underdrain, and surface drain. A surface-subsurface hydrologic model was implemented in the evaluation. The mathematical model incorporates processes of runoff inflow, direct precipitation, infiltration, evapotranspiration (ET), exfiltration, and outflow. This modeling assessment used a rain garden with subsoil permeability of 1 mm=h, ponding and media depth of 304.8 mm each, and loading ratio (contributory catchment area over rain garden area) of 5:1 under rainfalls over an entire year as well as individual design storms in New Jersey. Over an entire year, the longest continual ponding times are 880, 8, and 12 h for design alternatives of no drain, underdrain, and surface drain, respectively. Mosquito issues might arise from long ponding time for the no-drain design. Soil moisture of all three drainage designs never reaches down to the wilting point. Volume reduction of stormwater runoff for the three drainage designs are 89%, 15%, and 58%, respectively. The hydrologic performance for individual design storms (1, 2, 5, 10, 25, 50, and 100 year) was also addressed. Differences of effectiveness in runoff volume reduction for individual storms among the three drainage designs are consistent with those for the entire year. However, the underdrain design has the best performance in peak runoff rate reduction. Taken together, the surface drainage system has the best balance among runoff reduction for flood mitigation, soil moisture for plant survival, and ponding time for mosquito issues. The surface drainage design is recommended for a rain garden if it is installed on subsoil of low permeability and if the runoff captured does not require a substantial water quality improvement.

A graph G = (V, E) is called a split graph if there exists a partition V = I ∪K such that the subgraphs G[I] and G[K] of G induced by I and K are empty and complete graphs, respectively. In 1980, Burkard and Hammer gave a necessary condition for a split graph G with |I| < |K| to be hamiltonian. We will call a split graph G with |I| < |K| satisfying this condition a Burkard-Hammer graph. Further, a split graph G is called a maximal nonhamiltonian split graph if G is nonhamiltonian but G+uv is hamiltonian for every uv ∈ E where u ∈ I and v ∈ K. In an earlier work, the author and Iamjaroen have asked whether every maximal nonhamiltonian Burkard-Hammer graph G with the minimum degree δ(G) ≥ |I| − k where k ≥ 3 possesses a vertex adjacent to all vertices of G and whether every maximal nonhamiltonian Burkard-Hammer graph G with δ(G) = |I|−k where k > 3 and |I| > k +2 possesses a vertex with exactly k − 1 neighbors in I. The first question and the second one have been proved earlier to have a positive answer for k = 3 and k = 4, respectively. In this paper, we give a negative answer both to the first question for all k ≥ 4 and to the second question for all k ≥ 5.