On k-ordered Hamiltonian graphs

A Hamiltonian graph G of order n is k-ordered, 2 ≤ k ≤ n, if for every sequence v₁, v₂, …, v_k of k distinct vertices of G, there exists a Hamiltonian cycle that encounters v₁, v₂, …, v_k in this order. Define f(k, n) as the smallest integer m for which any graph on n vertices with minimum degree at least m is a k-ordered Hamiltonian graph. In this article, answering a question of Ng and Schultz, we determine f(k, n) if n is sufficiently large in terms of k. Let g(k, n) = − 1. More precisely, we show that f(k, n) = g(k, n) if n ≥ 11k − 3. Furthermore, we show that f(k, n) ≥ g(k, n) for any n ≥ 2k. Finally we show that f(k, n) > g(k, n) if 2k ≤ n ≤ 3k − 6. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 17–25, 1999     

Distance-biregular graphs with 2-valent vertices and distance-regular line graphs

A graph G is called distance-regularized if each vertex of G admits an intersection array. It is known that every distance-regularized graph is either distance-regular (DR) or distance-biregular (DBR). Note that DBR means that the graph is bipartite and the vertices in the same color class have the same intersection array. A (k, g)-graph is a k-regular graph with girth g and with the minimum possible number of vertices consistent with these properties. Biggs proved that, if the line graph L(G) is distance-transitive, then G is either K_1,n or a (k, g)-graph. This result is generalized to DR graphs by showing that the following are equivalent: (1) L(G) is DR and G ≠ K_1,n for n ≥ 2, (2) G and L(G) are both DR, (3) subdivision graph S(G) is DBR, and (4) G is a (k, g)-graph. This result is used to show that a graph S is a DBR graph with 2-valent vertices iff S = K_2,′ or S is the subdivision graph of a (k, g)-graph. Let G⁽²⁾ be the graph with vertex set that of G and two vertices adjacent if at distance two in G. It is shown that for a DBR graph G, G⁽²⁾ is two DR graphs. It is proved that a DR graph H without triangles can be obtained as a component of G⁽²⁾ if and only if it is a (k, g)-graph with g ≥ 4.     

On graphs satisfying a local ore-type condition

For an integer i, a graph is called an L_i-graph if, for each triple of vertices u, v, w with d(u, v) = 2 and w (element of) N(u) (intersection) N(v), d(u) + d(v) ≥ | N(u) (union) N(v) (union) N(w)| —i. Asratian and Khachatrian proved that connected L_o-graphs of order at least 3 are hamiltonian, thus improving Ore's Theorem. All K_1,3-free graphs are L₁-graphs, whence recognizing hamiltonian L₁-graphs is an NP-complete problem. The following results about L₁-graphs, unifying known results of Ore-type and known results on K_1,3-free graphs, are obtained. Set K = lcub;G|K_p,p+1 (contained within) G (contained within) K_p V K_p+1 for some ρ ≥ } (v denotes join). If G is a 2-connected L₁-graph, then G is 1-tough unless G (element of) K. Furthermore, if G is as connected L₁-graph of order at least 3 such that |N(u) (intersection) N(v)| ≥ 2 for every pair of vertices u, v with d(u, v) = 2, then G is hamiltonian unless G ϵ K, and every pair of vertices x, y with d(x, y) ≥ 3 is connected by a Hamilton path. This result implies that of Asratian and Khachatrian. Finally, if G is a connected L₁-graph of even order, then G has a perfect matching. © 1996 John Wiley & Sons, Inc.     

On the average genus of the random graph

The generalized Petersen graph GP (n, k), n ≤ 3, 1 ≥ k < n/2 is a cubic graph with vertex-set {u_j; i ? Z_n} ∪ {v_j; i ? Z_n}, and edge-set {u_iu_i, u_iv_i, v_iv_{i+k, i?}Z_n}. In the paper we prove that (i) GP(n, k) is a Cayley graph if and only if k² ? 1 (mod n); and (ii) GP(n, k) is a vertex-transitive graph that is not a Cayley graph if and only if k² ? -1 (mod n) or (n, k) = (10, 2), the exceptional graph being isomorphic to the 1-skeleton of the dodecahedon. The proof of (i) is based on the classification of orientable regular embeddings of the n-dipole, the graph consisting of two vertices and n parallel edges, while (ii) follows immediately from (i) and a result of R. Frucht, J.E. Graver, and M.E. Watkins [“The Groups of the Generalized Petersen Graphs,” Proceedings of the Cambridge Philosophical Society, Vol. 70 (1971), pp. 211-218]. © 1995 John Wiley & Sons, Inc.     

Linking (n − 2)-dimensional panels inn-space I: (k − 1,k)-graphs and (k − 1,k)-frames

A (k – 1,k)-graph is a multi-graph satisfyinge (k – 1)v – k for every non-empty subset ofe edges onv vertices, with equality whene = |E(G)|. A (k – 1,k)-frame is a structure generalizing an (n – 2, 2)-framework inn-space, a structure consisting of a set of (n – 2)-dimensional bodies inn-space and a set of rigid bars each joining a pair of bodies using ball joints. We prove that a graph is the graph of a minimally rigid (with respect to edges) (k – 1,k)-frame if and only if it is a (k – 1,k)-graph. Rigidity here means infinitesimal rigidity or equivalently statical rigidity.     

Hamiltonian paths and cycles in hypertournaments

Given two integers n and k, n ≥ k > 1, a k-hypertournament T on n vertices is a pair (V, A), where V is a set of vertices, |V| = n and A is a set of k-tuples of vertices, called arcs, so that for any k-subset S of V, A$ contains exactly one of the k! k-tuples whose entries belong to S. A 2-hypertournament is merely an (ordinary) tournament. A path is a sequence v₁a₁v₂v₃···v_t−1v_t of distinct vertices v₁, v₂,⋖, v_t and distinct arcs a₁, ⋖, a_t−1 such that v_i precedes v_t−1 in a, 1 ≤ i ≤ t − 1. A cycle can be defined analogously. A path or cycle containing all vertices of T (as v_i's) is Hamiltonian. T is strong if T has a path from x to y for every choice of distinct x, y ≡ V. We prove that every k-hypertournament on n (k) vertices has a Hamiltonian path (an extension of Redeis theorem on tournaments) and every strong k-hypertournament with n (k + 1) vertices has a Hamiltonian cycle (an extension of Camions theorem on tournaments). Despite the last result, it is shown that the Hamiltonian cycle problem remains polynomial time solvable only for k ≤ 3 and becomes NP-complete for every fixed integer k ≥ 4. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 277–286, 1997     

A note on a conjecture about cycles with many incident chords

Given positive integers n and k, let g_k(n) denote the maximum number of edges of a graph on n vertices that does not contain a cycle with k chords incident to a vertex on the cycle. Bollobás conjectured as an exercise in [2, p. 398, Problem 13] that there exists a function n(k) such that g_k(n) = (k + 1)n ? (k + 1)² for all n ≥ n(k). Using an old result of Bondy [ 3 ], we prove the conjecture, showing that n(k) ≤ 3 k + 3. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 180–182, 2004     

Perfect Matchings and K 4 3 -Tilings in Hypergraphs of Large Codegree

For a k-graph F, let t _l (n, m, F) be the smallest integer t such that every k-graph G on n vertices in which every l-set of vertices is included in at least t edges contains a collection of vertex-disjoint F-subgraphs covering all but at most m vertices of G. Let K _m ^k denote the complete k-graph on m vertices. The function $t_{k-1} (kn, 0, K_k^k)For a k-graph F, let t _l (n, m, F) be the smallest integer t such that every k-graph G on n vertices in which every l-set of vertices is included in at least t edges contains a collection of vertex-disjoint F-subgraphs covering all but at most m vertices of G. Let K _m ^k denote the complete k-graph on m vertices. The function (i.e. when we want to guarantee a perfect matching) has been previously determined by Kühn and Osthus [9] (asymptotically) and by R?dl, Ruciński, and Szemerédi [13] (exactly). Here we obtain asymptotic formulae for some other l. Namely, we prove that for any and , . Also, we present various bounds in another special but interesting case: t ₂(n, m, K ₄³) with m = 0 or m = o(n), that is, when we want to tile (almost) all vertices by copies of K ₄³, the complete 3-graph on 4 vertices. Reverts to public domain 28 years from publication. Oleg Pikhurko: Partially supported by the National Science Foundation, Grant DMS-0457512.     

Spanning trees with many leaves

Let a maximal chain of vertices of degree 2 in a graph G consist of k > 0 vertices. We prove that G has a spanning tree with more than \fracv(G)2k + 4 \frac{{v(G)}}{{2k + 4}} leaves (where υ(G) is the number of vertices of the graph G). We present an infinite series of examples showing that the constant \frac12k + 4 \frac{1}{{2k + 4}} cannot be enlarged. Bibliography: 7 titles.     

Bipartite labeling of trees with maximum degree three

Let T = (V, E) be a tree with a properly 2‐colored vertex set. A bipartite labeling of T is a bijection φ: V → {1, …, |V|} for which there exists a k such that whenever φ(u) ≤ k < φ(v), then u and v have different colors. The α‐size α(T) of the tree T is the maximum number of elements in the sets {|φ(u) − φ(v)|; uv ∈ E}, taken over all bipartite labelings φ of T. The quantity α(n) is defined as the minimum of α(T) over all trees with n vertices. In an earlier article (J Graph Theory 19 (1995), 201–215), A. Rosa and the second author proved that 5n/7 ≤ α(n) ≤ (5n + 4)/6 for all n ≥ 4; the upper bound is believed to be the asymptotically correct value of (n). In this article, we investigate the α‐size of trees with maximum degree three. Let α₃(n) be the smallest α‐size among all trees with n vertices, each of degree at most three. We prove that α₃(n) ≥ 5n/6 for all n ≥ 12, thus supporting the belief above. This result can be seen as an approximation toward the graceful tree conjecture—it shows that every tree on n ≥ 12 vertices and with maximum degree three has “gracesize” at least 5n/6. Using a computer search, we also establish that α₃(n) ≥ n − 2 for all n ≤ 17. © 1999 John Wiley & Sons, Inc. J Graph Theory 31:7–15, 1999     

Connectivity and separating sets of cages

A (k; g)-cage is a graph of minimum order among k-regular graphs with girth g. We show that for every cutset S of a (k; g)-cage G, the induced subgraph G[S] has diameter at least ⌊g/2⌋, with equality only when distance ⌊g/2⌋ occurs for at least two pairs of vertices in G[S]. This structural property is used to prove that every (k; g)-cage with k ≥ 3 is 3-connected. This result supports the conjecture of Fu, Huang, and Rodger that every (k; g)-cage is k-connected. A nonseparating g-cycle C in a graph G is a cycle of length g such that G − V(C) is connected. We prove that every (k; g)-cage contains a nonseparating g-cycle. For even g, we prove that every g-cycle in a (k; g)-cage is nonseparating. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 35–44, 1998     

On the Perfect Matchings of Near Regular Graphs

Let k, h be positive integers with k ≤ h. A graph G is called a [k, h]-graph if k ≤ d(v) ≤ h for any v ? V(G){v \in V(G)}. Let G be a [k, h]-graph of order 2n such that k ≥ n. Hilton (J. Graph Theory 9:193–196, 1985) proved that G contains at least ?k/3?{\lfloor k/3\rfloor} disjoint perfect matchings if h = k. Hilton’s result had been improved by Zhang and Zhu (J. Combin. Theory, Series B, 56:74–89, 1992), they proved that G contains at least ?k/2?{\lfloor k/2\rfloor} disjoint perfect matchings if k = h. In this paper, we improve Hilton’s result from another direction, we prove that Hilton’s result is true for [k, k + 1]-graphs. Specifically, we prove that G contains at least ?\fracn3?+1+(k-n){\lfloor\frac{n}3\rfloor+1+(k-n)} disjoint perfect matchings if h = k + 1.     

Mader’s Conjecture On Extremely Critical Graphs

A non-complete graph G is called an (n,k)-graph if it is n-connected but G—X is not (n−|X|+1)-connected for any X ⊂V (G) with |X|≤k. Mader conjectured that for k≥3 the graph K_2k+2−(1−factor) is the unique (2k,k)-graph(up to isomorphism). Here we prove this conjecture.     

Generalized Steiner systems GS4 (2, 4, v,g) for g = 2, 3, 6

Generalized Steiner systems GS_d(t, k, v, g) were first introduced by Etzion and used to construct optimal constant‐weight codes over an alphabet of size g + 1 with minimum Hamming distance d, in which each codeword has length v and weight k. Much work has been done for the existence of generalized Steiner triple systems GS(2, 3, v, g). However, for block size four there is not much known on GS_d(2, 4, v, g). In this paper, the necessary conditions for the existence of a GS_d(t, k, v, g) are given, which answers an open problem of Etzion. Some singular indirect product constructions for GS_d(2, k, v, g) are also presented. By using both recursive and direct constructions, it is proved that the necessary conditions for the existence of a GS₄(2, 4, v, g) are also sufficient for g = 2, 3, 6. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 401–423, 2001     

Long cycles passing through a specified path in a graph

For a graph G and an integer k ≥ 1, let ς_k(G) = d_G(v_i): {v₁, …, v_k} is an independent set of vertices in G}. Enomoto proved the following theorem. Let s ≥ 1 and let G be a (s + 2)-connected graph. Then G has a cycle of length ≥ min{|V(G)|, ς₂(G) − s} passing through any path of length s. We generalize this result as follows. Let k ≥ 3 and s ≥ 1 and let G be a (k + s − 1)-connected graph. Then G has a cycle of length ≥ min{|V(G)|, − s} passing through any path of length s. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 177–184, 1998     

Upper bound theorems for homology manifolds

In this paper we prove the Upper Bound Conjecture (UBC) for some classes of (simplicial) homology manifolds: we show that the UBC holds for all odd-dimensional homology manifolds and for all 2k-dimensional homology manifolds Δ such that β _k (Δ)⩽Σ{β _i (Δ):i ≠k-2,k,k+2 and 1 ⩽i⩽2k-1}, where β _i (Δ) are reduced Betti numbers of Δ. (This condition is satisfied by 2k-dimensional homology manifolds with Euler characteristic χ≤2 whenk is even or χ≥2 whenk is odd, and for those having vanishing middle homology.) We prove an analog of the UBC for all other even-dimensional homology manifolds. Kuhnel conjectured that for every 2k-dimensional combinatorial manifold withn vertices, . We prove this conjecture for all 2k-dimensional homology manifolds withn vertices, wheren≥4k+3 orn≤3k+3. We also obtain upper bounds on the (weighted) sum of the Betti numbers of odd-dimensional homology manifolds.     

Graphic Sequences Have Realizations Containing Bisections of Large Degree

A bisection of a graph is a balanced bipartite spanning sub‐graph. Bollobás and Scott conjectured that every graph G has a bisection H such that deg_H(v) ≥ ?deg_G(v)/2? for all vertices v. We prove a degree sequence version of this conjecture: given a graphic sequence π, we show that π has a realization G containing a bisection H where deg_H(v) ≥ ?(deg_G(v) ? 1)/2? for all vertices v. This bound is very close to best possible. We use this result to provide evidence for a conjecture of Brualdi (Colloq. Int. CNRS, vol. 260, CNRS, Paris) and Busch et al. (2011), that if π and π ? k are graphic sequences, then π has a realization containing k edge‐disjoint 1‐factors. We show that if the minimum entry δ in π is at least n/2 + 2, then π has a realization containing edge‐disjoint 1‐factors. We also give a construction showing the limits of our approach in proving this conjecture. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Upper and lower bounds of the valence-functional

For the non-negative integerg let (M, g) denote the closed orientable 2-dimensional manifold of genusg. K-realizationsP of (M, g) are geometric cell-complexes inP with convex facets such that set (P) is homeomorphic toM. ForK-realizationsP of (M, g) and verticesv ofP, val (v,P) denotes the number of edges ofP incident withv and the weighted vertex-number Σ(val(v, P)-3) taken over all vertices ofP is called valence-valuev (P) ofP. The valence-functionalV, which is important for the determination of all possiblef-vectors ofK-realisations of (M, g), in connection with Eberhard's problem etc., is defined byV(g):=min[v(P)|P is aK-realization of (M,g)]. The aim of the note is to prove the inequality 2g+1≦V(g)≦3g+3 for every positive integerg.     

Defective coloring revisited

A graph is (k, d)-colorable if one can color the vertices with k colors such that no vertex is adjacent to more than d vertices of its same color. In this paper we investigate the existence of such colorings in surfaces and the complexity of coloring problems. It is shown that a toroidal graph is (3, 2)- and (5, 1)-colorable, and that a graph of genus γ is (χ_γ/(d + 1) + 4, d)-colorable, where χ_γ is the maximum chromatic number of a graph embeddable on the surface of genus γ. It is shown that the (2, k)-coloring, for k ≥ 1, and the (3, 1)-coloring problems are NP-complete even for planar graphs. In general graphs (k, d)-coloring is NP-complete for k ≥ 3, d ≥ 0. The tightness is considered. Also, generalizations to defects of several algorithms for approximate (proper) coloring are presented. © 1997 John Wiley & Sons, Inc.