Centers to centroids in graphs

For S ? V(G) the S-center and S-centroid of G are defined as the collection of vertices u ∈ V(G) that minimize e_s(u) = max {d(u, v): v ∈ S} and d_s(u) = ∑_u∈S d(u, v), respectively. This generalizes the standard definition of center and centroid from the special case of S = V(G). For 1 ? k ?|V(G)| and u ∈ V(G) let r_k(u) = max {∑_s∈S d(u, s): S ? V(G), |S| = k}. The k-centrum of G, denoted C(G; k), is defined to be the subset of vertices u in G for which r_k(u) is a minimum. This also generalizes the standard definitions of center and centroid since C(G; 1) is the center and C(G; |V(G)|) is the centroid. In this paper the structure of these sets for trees is examined. Generalizations of theorems of Jordan and Zelinka are included.     

Partitions of a graph into paths with prescribed endvertices and lengths

For a graph G, let σ₂(G) denote the minimum degree sum of a pair of nonadjacent vertices. We conjecture that if |V(G)| = n = Σ^k_{i = 1} a_i and σ₂(G) ≥ n + k − 1, then for any k vertices v₁, v₂,…, v_k in G, there exist vertex‐disjoint paths P₁, P₂,…, P_k such that |V(P_i)| = a_i and v_i is an endvertex of P_i for 1 ≤ i ≤ k. In this paper, we verify the conjecture for the cases where almost all a_i ≤ 5, and the cases where k ≤ 3. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 163–169, 2000     

On hamiltonian-connected graphs

One of the most fundamental results concerning paths in graphs is due to Ore: In a graph G, if deg x + deg y ≧ |V(G)| + 1 for all pairs of nonadjacent vertices x, y ? V(G), then G is hamiltonian-connected. We generalize this result using set degrees. That is, for S ? V(G), let deg S = |_x?S N(x)|, where N(x) = {v|xv ? E(G)} is the neighborhood of x. In particular we show: In a 3-connected graph G, if deg S₁ + deg S₂ ≧ |V(G)| + 1 for each pair of distinct 2-sets of vertices S₁, S₂ ? V(G), then G is hamiltonian-connected. Several corollaries and related results are also discussed.     

List Colorings of K5‐Minor‐Free Graphs With Special List Assignments

The following question was raised by Bruce Richter. Let G be a planar, 3‐connected graph that is not a complete graph. Denoting by d(v) the degree of vertex v, is G L‐list colorable for every list assignment L with |L(v)| = min{d(v), 6} for all v∈V(G)? More generally, we ask for which pairs (r, k) the following question has an affirmative answer. Let r and k be the integers and let G be a K₅‐minor‐free r‐connected graph that is not a Gallai tree (i.e. at least one block of G is neither a complete graph nor an odd cycle). Is G L‐list colorable for every list assignment L with |L(v)| = min{d(v), k} for all v∈V(G)? We investigate this question by considering the components of G[S_k], where S_k: = {v∈V(G)|d(v)8k} is the set of vertices with small degree in G. We are especially interested in the minimum distance d(S_k) in G between the components of G[S_k]. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:18–30, 2012     

A 2-factor with Short Cycles Passing Through Specified Independent Vertices in Graph

For a graph G, we define σ₂(G) := min{d(u) + d(v)|u, v ≠ ∈ E(G), u ≠ v}. Let k ≥ 1 be an integer and G be a graph of order n ≥ 3k. We prove if σ₂(G) ≥ n + k − 1, then for any set of k independent vertices v ₁,...,v _k , G has k vertex-disjoint cycles C ₁,..., C _k of length at most four such that v _i ∈ V(C _i ) for all 1 ≤ i ≤ k. And show if σ₂(G) ≥ n + k − 1, then for any set of k independent vertices v ₁,...,v _k , G has k vertex-disjoint cycles C ₁,..., C _k such that v _i ∈ V(C _i ) for all 1 ≤ i ≤ k, V(C ₁) ∪...∪ V(C _k ) = V(G), and |C _i | ≤ 4 for all 1 ≤ i ≤ k − 1. The condition of degree sum σ₂(G) ≥ n + k − 1 is sharp. Received: December 20, 2006. Final version received: December 12, 2007.     

Independent edges in bipartite graphs obtained from orientations of graphs

Given a digraph D on vertices v₁, v₂, ?, v_n, we can associate a bipartite graph B(D) on vertices s₁, s₂, ?, s_n, t₁, t₂, ?, t_n, where s_it_j is an edge of B(D) if (v_i, v_j) is an arc in D. Let O_G denote the set of all orientations on the (undirected) graph G. In this paper we will discuss properties of the set S(G) := {β₁ (B(D))) | D ? O_G}, where β₁ is the edge independence number. In the first section we present some background and related concepts. We show that sets of the form S(G) are convex and that max S(G) ≦ 2 min S(G). Furthermore, this completely characterizes such sets. In the second section we discuss some bounds on elements of S(G) in terms of more familiar graphical parameters. The third section deals with extremal problems. We discuss bounds on elements of S(G) if the order and size of G are known, particularly when G is bipartite. In this section we exhibit a relation between max S(G) and the concept of graphical closure. In the fourth and final section we discuss the computational complexity of computing min S(G) and max S(G). We show that the first problem is NP-complete and that the latter is polynomial.     

A Remark on Hamiltonian Cycles

Let G be an undirected and simple graph on n vertices. Let ω, α and χ denote the number of components, the independence number and the connectivity number of G. G is called a 1-tough graph if ω(G – S) ? |S| for any subset S of V(G) such that ω(G ? S) > 1. Let σ₂ = min {d(v) + d(w)|v and w are nonadjacent}. Note that the difference α - χ in 1-tough graph may be made arbitrary large. In this paper we prove that any 1-tough graph with σ₂ > n + χ - α is hamiltonian.     

Partition conditions and vertex-connectivity of graphs

It was proved ([5], [6]) that ifG is ann-vertex-connected graph then for any vertex sequencev ₁, ...,v _n ≠V(G) and for any sequence of positive integersk ₁, ...,k _n such thatk ₁+...+k _n =|V(G)|, there exists ann-partition ofV(G) such that this partition separates the verticesv ₁, ...,v(n), and the class of the partition containingv _i induces a connected subgraph consisting ofk _i vertices, fori=1, 2, ...,n. Now fix the integersk ₁, ...,k _n . In this paper we study what can we say about the vertex-connectivity ofG if there exists such a partition ofV(G) for any sequence of verticesv ₁, ...,v _n ≠V(G). We find some interesting cases when the existence of such partitions implies then-vertex-connectivity ofG, in the other cases we give sharp lower bounds for the vertex-connectivity ofG.     

Cycles containing all vertices of maximum degree

For a graph G and an integer k, denote by V_k the set {v ∈ V(G) | d(v) ≥ k}. Veldman proved that if G is a 2-connected graph of order n with n ≤ 3k - 2 and |V_k| ≤ k, then G has a cycle containing all vertices of V_k. It is shown that the upper bound k on |V_k| is close to best possible in general. For the special case k = δ(G), it is conjectured that the condition |V_k| ≤ k can be omitted. Using a variation of Woodall's Hopping Lemma, the conjecture is proved under the additional condition that n ≤ 2δ(G) + δ(G) + 1. This result is an almost-generalization of Jackson's Theorem that every 2-connected k-regular graph of order n with n ≤ 3k is hamiltonian. An alternative proof of an extension of Jackson's Theorem is also presented. © 1993 John Wiley & Sons, Inc.     

Mutually Independent Hamiltonian Connectivity of (n, k)-Star Graphs

A graph G is hamiltonian connected if there exists a hamiltonian path joining any two distinct nodes of G. Two hamiltonian paths and of G from u to v are independent if u = u ₁ = v ₁, v = u _v(G) = v _v(G) , and u _i ≠ v _i for every 1 < i < v(G). A set of hamiltonian paths, {P ₁, P ₂, . . . , P _k }, of G from u to v are mutually independent if any two different hamiltonian paths are independent from u to v. A graph is k mutually independent hamiltonian connected if for any two distinct nodes u and v, there are k mutually independent hamiltonian paths from u to v. The mutually independent hamiltonian connectivity of a graph G, IHP(G), is the maximum integer k such that G is k mutually independent hamiltonian connected. Let n and k be any two distinct positive integers with n–k ≥ 2. We use S _n,k to denote the (n, k)-star graph. In this paper, we prove that IHP(S _n,k ) = n–2 except for S _4,2 such that IHP(S _4,2) = 1.      

Acyclic 6-choosability of planar graphs without adjacent short cycles

A proper vertex coloring of a graph G is acyclic if G contains no bicolored cycles.Given a list assignment L={L(v)|v∈V}of G,we say that G is acyclically L-colorable if there exists a proper acyclic coloringπof G such thatπ(v)∈L(v)for all v∈V.If G is acyclically L-colorable for any list assignment L with|L(v)|k for all v∈V(G),then G is acyclically k-choosable.In this paper,we prove that every planar graph G is acyclically 6-choosable if G does not contain 4-cycles adjacent to i-cycles for each i∈{3,4,5,6}.This improves the result by Wang and Chen(2009).     

Some results on generalized exponents

A digraph G = (V, E) is primitive if, for some positive integer k, there is a u → v walk of length k for every pair u, v of vertices of V. The minimum such k is called the exponent of G, denoted exp(G). The exponent of a vertex u ∈ V, denoted exp(u), is the least integer k such that there is a u → v walk of length k for each v ∈ V. For a set X ⊆ V, exp(X) is the least integer k such that for each v ∈ V there is a X → v walk of length k, i.e., a u → v walk of length k for some u ∈ X. Let F(G, k) : = max{exp(X) : |X| = k} and F(n, k) : = max{F(G, k) : |V| = n}, where |X| and |V| denote the number of vertices in X and V, respectively. Recently, B. Liu and Q. Li proved F(n, k) = (n − k)(n − 1) + 1 for all 1 ≤ k ≤ n − 1. In this article, for each k, 1 ≤ k ≤ n − 1, we characterize the digraphs G such that F(G, k) = F(n, k), thereby answering a question of R. Brualdi and B. Liu. We also find some new upper bounds on the (ordinary) exponent of G in terms of the maximum outdegree of G, Δ⁺(G) = max{d⁺(u) : u ∈ V}, and thus obtain a new refinement of the Wielandt bound (n − 1)² + 1. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 215–225, 1998     

On H-Linked Graphs

For a fixed multigraph H, possibly containing loops, with V(H)={h₁, . . . , h_k}, we say a graph G is H-linked if for every choice of k vertices v₁, . . . , v_k in G, there exists a subdivision of H in G such that v_i represents h_i (for all i). This notion clearly generalizes the concept of k-linked graphs (as well as other properties). In this paper we determine, for a connected multigraph H and for any sufficiently large graph G, a sharp lower bound on δ(G) (depending upon H) such that G is H-linked.     

Acyclic 4-choosability of planar graphs

A proper vertex coloring of a graph G=(V,E) is acyclic if G contains no bicolored cycle. Given a list assignment L={L(v)∣v∈V} of G, we say G is acyclically L-list colorable if there exists a proper acyclic coloring π of G such that π(v)∈L(v) for all v∈V. If G is acyclically L-list colorable for any list assignment with |L(v)|≥k for all v∈V, then G is acyclically k-choosable. In this paper we prove that planar graphs without 4, 7, and 8-cycles are acyclically 4-choosable.     

Spanning trees with leaf distance at least four

For a graph G, we denote by i(G) the number of isolated vertices of G. We prove that for a connected graph G of order at least five, if i(G–S) < |S| for all ?? ≠ S ? V(G), then G has a spanning tree T such that the distance in T between any two leaves of T is at least four. This result was conjectured by Kaneko in “Spanning trees with constrains on the leaf degree”, Discrete Applied Math, 115 (2001), 73–76. Moreover, the condition in the result is sharp in a sense that the condition i(G–S) < |S| cannot be replaced by i(G–S) ≤ |S|. © 2006 Wiley Periodicals, Inc. J Graph Theory 55: 83–90, 2007     

Competitively Tight Graphs

The competition graph of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if there exists a vertex v in D such that (x, v) and (y, v) are arcs of D. For any graph G, G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number k(G) of a graph G is the smallest number of such isolated vertices. Computing the competition number of a graph is an NP-hard problem in general and has been one of the important research problems in the study of competition graphs. Opsut [1982] showed that the competition number of a graph G is related to the edge clique cover number θ _E (G) of the graph G via θ _E (G) ? |V(G)| + 2 ≤ k(G) ≤ θ _E (G). We first show that for any positive integer m satisfying 2 ≤ m ≤ |V(G)|, there exists a graph G with k(G) = θ _E (G) ? |V(G)| + m and characterize a graph G satisfying k(G) = θ _E (G). We then focus on what we call competitively tight graphs G which satisfy the lower bound, i.e., k(G) = θ _E (G) ? |V(G)| + 2. We completely characterize the competitively tight graphs having at most two triangles. In addition, we provide a new upper bound for the competition number of a graph from which we derive a sufficient condition and a necessary condition for a graph to be competitively tight.     

An Introduction to Closed/Open Neighborhood Sums: Minimax, Maximin, and Spread

For a graph G of order |V(G)| = n and a real-valued mapping f:V(G)?\mathbbR{f:V(G)\rightarrow\mathbb{R}}, if S ì V(G){S\subset V(G)} then f(S)=?_{w ? S} f(w){f(S)=\sum_{w\in S} f(w)} is called the weight of S under f. The closed (respectively, open) neighborhood sum of f is the maximum weight of a closed (respectively, open) neighborhood under f, that is, NS[f]=max{f(N[v])|v ? V(G)}{NS[f]={\rm max}\{f(N[v])|v \in V(G)\}} and NS(f)=max{f(N(v))|v ? V(G)}{NS(f)={\rm max}\{f(N(v))|v \in V(G)\}}. The closed (respectively, open) lower neighborhood sum of f is the minimum weight of a closed (respectively, open) neighborhood under f, that is, NS^-[f]=min{f(N[v])|v ? V(G)}{NS^{-}[f]={\rm min}\{f(N[v])|v\in V(G)\}} and NS^-(f)=min{f(N(v))|v ? V(G)}{NS^{-}(f)={\rm min}\{f(N(v))|v\in V(G)\}}. For W ì \mathbbR{W\subset \mathbb{R}}, the closed and open neighborhood sum parameters are NS_W[G]=min{NS[f]|f:V(G)? W{NS_W[G]={\rm min}\{NS[f]|f:V(G)\rightarrow W} is a bijection} and NS_W(G)=min{NS(f)|f:V(G)? W{NS_W(G)={\rm min}\{NS(f)|f:V(G)\rightarrow W} is a bijection}. The lower neighbor sum parameters are NS^-_W[G]=maxNS^-[f]|f:V(G)? W{NS^{-}_W[G]={\rm max}NS^{-}[f]|f:V(G)\rightarrow W} is a bijection} and NS^-_W(G)=maxNS^-(f)|f:V(G)? W{NS^{-}_W(G)={\rm max}NS^{-}(f)|f:V(G)\rightarrow W} is a bijection}. For bijections f:V(G)? {1,2,?,n}{f:V(G)\rightarrow \{1,2,\ldots,n\}} we consider the parameters NS[G], NS(G), NS ⁻[G] and NS ⁻(G), as well as two parameters minimizing the maximum difference in neighborhood sums.     

The partition dimension of circulant graphs

Let Π = {S₁, S₂, . . . , S_k} be an ordered partition of the vertex set V (G) of a graph G. The partition representation of a vertex v ∈ V (G) with respect to Π is the k-tuple r(v|Π) = (d(v, S₁), d(v, S₂), . . . , d(v, S_k)), where d(v, S) is the distance between v and a set S. If for every pair of distinct vertices u, v ∈ V (G), we have r(u|Π) ≠ r(v|Π), then Π is a resolving partition and the minimum cardinality of a resolving partition of V (G) is called the partition dimension of G. We study the partition dimension of circulant graphs, which are Cayley graphs of cyclic groups. Grigorious et al. [On the partition dimension of circulant graphs] proved that pd(C_n(1, 2, . . . , t)) ≥ t + 1 for n ≥ 3. We disprove this statement by showing that if t ≥ 4 is even, then there exists an infinite set of values of n, such that . We also present exact values of the partition dimension of circulant graphs with 3 generators.     

Acyclic 5‐choosability of planar graphs without small cycles

A proper vertex coloring of a graph G = (V,E) is acyclic if G contains no bicolored cycle. A graph G is acyclically L‐list colorable if for a given list assignment L = {L(v): v: ∈ V}, there exists a proper acyclic coloring ? of G such that ?(v) ∈ L(v) for all v ∈ V. If G is acyclically L‐list colorable for any list assignment with |L (v)|≥ k for all v ∈ V, then G is acyclically k‐choosable. In this article, we prove that every planar graph G without 4‐ and 5‐cycles, or without 4‐ and 6‐cycles is acyclically 5‐choosable. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 245–260, 2007