A Note on L（2, 1）-labelling of Trees

An L(2,1)-labelling of a graph G is a function from the vertex set V (G) to the set of all nonnegative integers such that |f(u) f(v)| ≥ 2 if d G (u,v)=1 and |f(u) f(v)| ≥ 1 if d G (u,v)=2.The L(2,1)-labelling problem is to find the smallest number,denoted by λ(G),such that there exists an L(2,1)-labelling function with no label greater than it.In this paper,we study this problem for trees.Our results improve the result of Wang [The L(2,1)-labelling of trees,Discrete Appl.Math.154 (2006) 598-603].     

On distance two graphs of upper bound graphs

For a poset P=(X,≤), the upper bound graph (UB-graph) of P is the graph U=(X,E_U), where uv∈E_U if and only if u≠v and there exists m∈X such that u,v≤m. For a graph G, the distance two graph DS₂(G) is the graph with vertex set V(DS₂(G))=V(G) and u,v∈V(DS₂(G)) are adjacent if and only if d_G(u,v)=2. In this paper, we deal with distance two graphs of upper bound graphs. We obtain a characterization of distance two graphs of split upper bound graphs.     

Hamiltonicity of 6-connected line graphs

Let G be a graph and let D₆(G)={v∈V(G)|d_G(v)=6}. In this paper we prove that: (i) If G is a 6-connected claw-free graph and if |D₆(G)|≤74 or G[D₆(G)] contains at most 8 vertex disjoint K₄’s, then G is Hamiltonian; (ii) If G is a 6-connected line graph and if |D₆(G)|≤54 or G[D₆(G)] contains at most 5 vertex disjoint K₄’s, then G is Hamilton-connected.     

A distance approximating trees

A 1-approximation of connected graph G=(V,E) is a tree T=(V,E^′) with the same vertex set such that for every two vertices |d_G(u,v)−d_T(u,v)|1. A polynomial time algorithm is designed for finding such a tree.     

Degree bounds for the circumference of 3‐connected graphs

Let C be a longest cycle in the 3‐connected graph G and let H be a component of G ? V(C) such that |V(H)| ≥ 3. We supply estimates of the form |C| ≥ 2d(u) + 2d(v) ? α(4 ≤ α ≤ 8), where u,v are suitably chosen non‐adjacent vertices in G. Also the exceptional classes for α = 6,7,8 are characterized. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Brooks-type theorems for choosability with separation

We consider the following type of problems. Given a graph G = (V, E) and lists L(v) of allowed colors for its vertices v ∈ V such that |L(v)| = p for all v ∈ V and |L(u) ∩ L(v)| ≤ c for all uv ∈ E, is it possible to find a “list coloring,” i.e., a color f(v) ∈ L(v) for each v ∈ V, so that f(u) ≠ f(v) for all uv ∈ E? We prove that every of maximum degree Δ admits a list coloring for every such list assignment, provided p ≥ . Apart from a multiplicative constant, the result is tight, as lists of length may be necessary. Moreover, for G = K_n (the complete graph on n vertices) and c = 1 (i.e., almost disjoint lists), the smallest value of p is shown to have asymptotics (1 + o(1)) . For planar graphs and c = 1, lists of length 4 suffice. ˜© 1998 John Wiley & Sons, Inc. J Graph Theory 27: 43–49, 1998     

Bi‐arc graphs and the complexity of list homomorphisms

Given graphs G, H, and lists L(v) ? V(H), v ε V(G), a list homomorphism of G to H with respect to the lists L is a mapping f : V(G) → V(H) such that uv ε E(G) implies f(u)f(v) ε E(H), and f(v) ε L(v) for all v ε V(G). The list homomorphism problem for a fixed graph H asks whether or not an input graph G, together with lists L(v) ? V(H), v ε V(G), admits a list homomorphism with respect to L. In two earlier papers, we classified the complexity of the list homomorphism problem in two important special cases: When H is a reflexive graph (every vertex has a loop), the problem is polynomial time solvable if H is an interval graph, and is NP‐complete otherwise. When H is an irreflexive graph (no vertex has a loop), the problem is polynomial time solvable if H is bipartite and H is a circular arc graph, and is NP‐complete otherwise. In this paper, we extend these classifications to arbitrary graphs H (each vertex may or may not have a loop). We introduce a new class of graphs, called bi‐arc graphs, which contains both reflexive interval graphs (and no other reflexive graphs), and bipartite graphs with circular arc complements (and no other irreflexive graphs). We show that the problem is polynomial time solvable when H is a bi‐arc graph, and is NP‐complete otherwise. In the case when H is a tree (with loops allowed), we give a simpler algorithm based on a structural characterization. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 61–80, 2003     

Independent sets in tensor graph powers

The tensor product of two graphs, G and H, has a vertex set V(G) × V(H) and an edge between (u,v) and (u′,v′) iff both u u′ ∈ E(G) and v v′ ∈ E(H). Let A(G) denote the limit of the independence ratios of tensor powers of G, lim, α(Gⁿ)/|V(Gⁿ)|. This parameter was introduced in [Brown, Nowakowski, Rall, SIAM J Discrete Math 9 ( 5 ), 290–300], where it was shown that A(G) is lower bounded by the vertex expansion ratio of independent sets of G. In this article we study the relation between these parameters further, and ask whether they are in fact equal. We present several families of graphs where equality holds, and discuss the effect the above question has on various open problems related to tensor graph products. © 2006 Wiley Periodicals, Inc. J Graph Theory     

A generalization of Fan’s results: Distribution of cycle lengths in graphs

Fan [G. Fan, Distribution of cycle lengths in graphs, J. Combin. Theory Ser. B 84 (2002) 187-202] proved that if G is a graph with minimum degree δ(G)≥3k for any positive integer k, then G contains k+1 cycles C₀,C₁,…,C_k such that k+1<|E(C₀)|<|E(C₁)|<?<|E(C_k)|, |E(C_i)−E(C_i−1)|=2, 1≤i≤k−1, and 1≤|E(C_k)|−|E(C_k−1)|≤2, and furthermore, if δ(G)≥3k+1, then |E(C_k)|−|E(C_k−1)|=2. In this paper, we generalize Fan’s result, and show that if we let G be a graph with minimum degree δ(G)≥3, for any positive integer k (if k≥2, then δ(G)≥4), if d_G(u)+d_G(v)≥6k−1 for every pair of adjacent vertices u,v∈V(G), then G contains k+1 cycles C₀,C₁,…,C_k such that k+1<|E(C₀)|<|E(C₁)|<?<|E(C_k)|, |E(C_i)−E(C_i−1)|=2, 1≤i≤k−1, and 1≤|E(C_k)|−|E(C_k−1)|≤2, and furthermore, if d_G(u)+d_G(v)≥6k+1, then |E(C_k)|−|E(C_k−1)|=2.     

The signed star domination numbers of the Cartesian product graphs

Let G be a graph with vertex set V(G) and edge set E(G). A function f:E(G)→{-1,1} is said to be a signed star dominating function of G if for every v∈V(G), where E_G(v)={uv∈E(G)|u∈V(G)}. The minimum of the values of , taken over all signed star dominating functions f on G, is called the signed star domination number of G and is denoted by γ_SS(G). In this paper, a sharp upper bound of γ_SS(G×H) is presented.     

Pan-connectedness of graphs with large neighborhood unions

Let G be a simple graph with n vertices. For any v ? V(G){v \in V(G)} , let N(v)={u ? V(G): uv ? E(G)}{N(v)=\{u \in V(G): uv \in E(G)\}} , NC(G) = min{|N(u) èN(v)|: u, v ? V(G){NC(G)= \min \{|N(u) \cup N(v)|: u, v \in V(G)} and uv \not ? E(G)}{uv \not \in E(G)\}} , and NC₂(G) = min{|N(u) èN(v)|: u, v ? V(G){NC_2(G)= \min\{|N(u) \cup N(v)|: u, v \in V(G)} and u and v has distance 2 in E(G)}. Let l ≥ 1 be an integer. A graph G on n ≥ l vertices is [l, n]-pan-connected if for any u, v ? V(G){u, v \in V(G)} , and any integer m with l ≤ m ≤ n, G has a (u, v)-path of length m. In 1998, Wei and Zhu (Graphs Combinatorics 14:263–274, 1998) proved that for a three-connected graph on n ≥ 7 vertices, if NC(G) ≥ n − δ(G) + 1, then G is [6, n]-pan-connected. They conjectured that such graphs should be [5, n]-pan-connected. In this paper, we prove that for a three-connected graph on n ≥ 7 vertices, if NC ₂(G) ≥ n − δ(G) + 1, then G is [5, n]-pan-connected. Consequently, the conjecture of Wei and Zhu is proved as NC ₂(G) ≥ NC(G). Furthermore, we show that the lower bound is best possible and characterize all 2-connected graphs with NC ₂(G) ≥ n − δ(G) + 1 which are not [4, n]-pan-connected.     

On Solvability of Finite Groups with Few Non-Normal Subgroups

For a finite group G, let v(G) denote the number of conjugacy classes of non-normal subgroups of G and vc(G) denote the number of conjugacy classes of non-normal noncyclic subgroups of G. In this paper, we show that every finite group G satisfying v(G) ≤2|π(G)| or vc(G) ≤ |π(G)| is solvable, and for a finite nonsolvable group G, v(G) = 2|π(G)| +1 if and only if G ? A ₅.     

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.     

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.     

Induced Matching Extendable Graph Powers

A graph G is called induced matching extendable (shortly, IM-extendable) if every induced matching of G is included in a perfect matching of G. A graph G is called strongly IM-extendable if every spanning supergraph of G is IM-extendable. The k-th power of a graph G, denoted by G^k, is the graph with vertex set V(G) in which two vertices are adjacent if and only if the distance between them in G is at most k. We obtain the following two results which give positive answers to two conjectures of Yuan. Result 1. If a connected graph G with |V(G)| even is locally connected, then G² is strongly IM-extendable. Result 2. If G is a 2-connected graph with |V(G)| even, then G³ is strongly IM-extendable. Research Supported by NSFC Fund 10371102.     

Acyclic 5-choosability of planar graphs without 4-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 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 every planar graph without 4-cycles and without two 3-cycles at distance less than 3 is acyclically 5-choosable. This improves a result in [M. Montassier, P. Ochem, A. Raspaud, On the acyclic choosability of graphs, J. Graph Theory 51 (2006) 281-300], which says that planar graphs of girth at least 5 are acyclically 5-choosable.     

Long paths through four vertices in a 2‐connected graph

Let G be a 2‐connected graph, let u and v be distinct vertices in V(G), and let X be a set of at most four vertices lying on a common (u, v)‐path in G. If deg(x) ≥ d for all x ∈ V(G) \ {u, v}, then there is a (u, v)‐path P in G with X ⊂ V(P) and |E(P)| ≥ d. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 55–65, 2000     

Decomposing large graphs with small graphs of high density

It is shown that for every positive integer h, and for every ϵ > 0, there are graphs H = )V_H E_H) with at least h vertices and with density at least 0.5 - ϵ with the following property: If G = (V_G, E_G) is any graph with minimum degree at least (1 + o(1)) and |E_H| divides |E_G|, then G has an H-decomposition. This result extends the results of [R. M. Wilson, Cong Numer XV (1925), 647–659] [T. Gustavsson, Ph.D. Thesis, U. Stockholm, 1991] [R. Yuster, Random Struc Algorith, 12 (1998), 237–251]. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 27–40, 1999     

A characterization of panconnected graphs satisfying a local ore-type condition

It is well known that a graph G of order p ≥ 3 is Hamilton-connected if d(u) + d(v) ≥ p + 1 for each pair of nonadjacent vertices u and v. In this paper we consider connected graphs G of order at least 3 for which d(u) + d(v) ≥ |N(u) ∪ N(v) ∪ N(w)| + 1 for any path uwv with uv ∉ E(G), where N(x) denote the neighborhood of a vertex x. We prove that a graph G satisfying this condition has the following properties: (a) For each pair of nonadjacent vertices x, y of G and for each integer k, d(x, y) ≤ k ≤ |V(G)| − 1, there is an x − y path of length k. (b) For each edge xy of G and for each integer k (excepting maybe one k η {3,4}) there is a cycle of length k containing xy. Consequently G is panconnected (and also edge pancyclic) if and only if each edge of G belongs to a triangle and a quadrangle. Our results imply some results of Williamson, Faudree, and Schelp. © 1996 John Wiley & Sons, Inc.     

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.