Local and global average degree in graphs and multigraphs

A vertex v of a graph G is called groupie if the average degree t_v of all neighbors of v in G is not smaller than the average degree t_G of G. Every graph contains a groupie vertex; the problem of whether or not every simple graph on ≧2 vertices has at least two groupie vertices turned out to be surprisingly difficult. We present various sufficient conditions for a simple graph to contain at least two groupie vertices. Further, we investigate the function f(n) = max min_v (t_v/t_G), where the maximum ranges over all simple graphs on n vertices, and prove that f(n) = 1/42n + o(1). The corresponding result for multigraphs is in sharp contrast with the above. We also characterize trees in which the local average degree t_v is constant.     

Roman and Total Domination

Abstract

A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set is the total domination number ^γt(G). A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f (u)=0 is adjacent to at least one vertex v of G for which f (v)=2. The minimum of f (V (G))=∑_{u ∈ V (G)} f (u) over all such functions is called the Roman domination number γR (G). We show that γt(G) ≤ γR (G) with equality if and only if ^γt(G)=2γ(G), where γ(G) is the domination number of G. Moreover, we characterize the extremal graphs for some graph families.     

Circular consecutive choosability of k‐choosable graphs

Let S(r) denote a circle of circumference r. The circular consecutive choosability ch_cc(G) of a graph G is the least real number t such that for any r≥χ_c(G), if each vertex v is assigned a closed interval L(v) of length t on S(r), then there is a circular r‐coloring f of G such that f(v)∈L(v). We investigate, for a graph, the relations between its circular consecutive choosability and choosability. It is proved that for any positive integer k, if a graph G is k‐choosable, then ch_cc(G)?k + 1 ? 1/k; moreover, the bound is sharp for k≥3. For k = 2, it is proved that if G is 2‐choosable then ch_cc(G)?2, while the equality holds if and only if G contains a cycle. In addition, we prove that there exist circular consecutive 2‐choosable graphs which are not 2‐choosable. In particular, it is shown that ch_cc(G) = 2 holds for all cycles and for K_{2, n} with n≥2. On the other hand, we prove that ch_cc(G)>2 holds for many generalized theta graphs. © 2011 Wiley Periodicals, Inc. J Graph Theory 67: 178‐197, 2011     

Up-embeddability via girth and the degree-sum of adjacent vertices

Let G be a simple graph of order n and girth g. For any two adjacent vertices u and v of G, if d _G (u) + d _G (v) ⩾ n − 2g + 5 then G is up-embeddable. In the case of 2-edge-connected (resp. 3-edge-connected) graph, G is up-embeddable if d _G (u) + d _G (v) ⩾ n − 2g + 3 (resp. d _G (u) + d _G (v) ⩾ n − 2g −5) for any two adjacent vertices u and v of G. Furthermore, the above three lower bounds are all shown to be tight. This work was supported by National Natural Science Foundation of China (Grant No. 10571013)     

Neighborhood unions and hamilton cycles

Let G be a graph on n vertices and N₂(G) denote the minimum size of N(u) ∪ N(v) taken over all pairs of independent vertices u, v of G. We show that if G is 3-connected and N₂(G) ? ½(n + 1), then G has a Hamilton cycle. We show further that if G is 2-connected and N₂(G) ? ½(n + 3), then either G has a Hamilton cycle or else G belongs to one of three families of exceptional graphs.     

Random lifts of graphs: Independence and chromatic number

For a graph G, a random n‐lift of G has the vertex set V(G)×[n] and for each edge [u, v] ∈ E(G), there is a random matching between {u}×[n] and {v}×[n]. We present bounds on the chromatic number and on the independence number of typical random lifts, with G fixed and n→∞. For the independence number, upper and lower bounds are obtained as solutions to certain optimization problems on the base graph. For a base graph G with chromatic number χ and fractional chromatic number χ_f, we show that the chromatic number of typical lifts is bounded from below by const ? and also by const ? χ_f/log²χ_f (trivially, it is bounded by χ from above). We have examples of graphs where the chromatic number of the lift equals χ almost surely, and others where it is a.s. O(χ/logχ). Many interesting problems remain open. © 2002 John Wiley & Sons, Inc. Random Struct. Alg., 20, 1–22, 2002     

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 integral sum graphs with a saturated vertex

As introduced by F.Harary in 1994, a graph G is said to be an integral sum graph if its vertices can be given a labeling f with distinct integers so that for any two distinct vertices u and v of G, uv is an edge of G if and only if f(u)+f(v) = f(w) for some vertex w in G.     

Lollipop graphs are extremal for commute times

Consider a simple random walk on a connected graph G=(V, E). Let C(u, v) be the expected time taken for the walk starting at vertex u to reach vertex v and then go back to u again, i.e., the commute time for u and v, and let C(G)=max_{u, v∈V}C(u, v). Further, let 𝒢(n, m) be the family of connected graphs on n vertices with m edges, , and let 𝒢(n)=∪_m𝒢(n, m) be the family of all connected n‐vertex graphs. It is proved that if G∈(n, m) is such that C(G)=max_{H∈𝒢(n, m)}C(H) then G is either a lollipop graph or a so‐called double‐handled lollipop graph. It is further shown, using this result, that if C(G)=max_H∈𝒢(n)C(H) then G is the full lollipop graph or a full double‐handled lollipop graph with [(2n−1)/3] vertices in the clique unless n≤9 in which case G is the n‐path. ©2000 John Wiley & Sons, Inc. Random Struct. Alg., 16, 131–142, 2000     

The 2-pebbling property for dense graphs

Given a distribution of pebbles on the vertices of a connected graph G, a pebbling move on G consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The pebbling number f(G) is the smallest number m such that for every distribution of m pebbles and every vertex v,a pebble can be moved to v. A graph G is said to have the 2-pebbling property if for any distribution with more than 2f(G) q pebbles, where q is the number of vertices with at least one pebble, it is possible,using pebbling moves, to get two pebbles to any vertex. Snevily conjectured that G(s,t) has the 2-pebbling property, where G(s, t) is a bipartite graph with partite sets of size s and t (s ≥ t). Similarly, the-pebbling number f (G) is the smallest number m such that for every distribution of m pebbles and every vertex v, pebbles can be moved to v. Herscovici et al. conjectured that f(G) ≤ 1.5n + 8-6 for the graph G with diameter 3, where n = |V (G)|. In this paper, we prove that if s ≥ 15 and G(s, t) has minimum degree at least (s+1)/ 2 , then f (G(s, t)) = s + t, G(s, t) has the 2-pebbling property and f (G(s, t)) ≤ s + t + 8(-1). In other words, we extend a result due to Czygrinow and Hurlbert, and show that the above Snevily conjecture and Herscovici et al. conjecture are true for G(s, t) with s ≥ 15 and minimum degree at least (s+1)/ 2 .     

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.     

The upper traceable number of a graph

For a nontrivial connected graph G of order n and a linear ordering s: v ₁, v ₂, …, v _n of vertices of G, define . The traceable number t(G) of a graph G is t(G) = min{d(s)} and the upper traceable number t ⁺(G) of G is t ⁺(G) = max{d(s)}, where the minimum and maximum are taken over all linear orderings s of vertices of G. We study upper traceable numbers of several classes of graphs and the relationship between the traceable number and upper traceable number of a graph. All connected graphs G for which t ⁺(G) − t(G) = 1 are characterized and a formula for the upper traceable number of a tree is established. Research supported by Srinakharinwirot University, the Thailand Research Fund and the Commission on Higher Education, Thailand under the grant number MRG 5080075.     

On circuits and pancyclic line graphs

Clark proved that L(G) is hamiltonian if G is a connected graph of order n ≥ 6 such that deg u + deg v ≥ n – 1 – p(n) for every edge uv of G, where p(n) = 0 if n is even and p(n) = 1 if n is odd. Here it is shown that the bound n – 1 – p(n) can be decreased to (2n + 1)/3 if every bridge of G is incident with a vertex of degree 1, which is a necessary condition for hamiltonicity of L(G). Moreover, the conclusion that L(G) is hamiltonian can be strengthened to the conclusion that L(G) is pancyclic. Lesniak-Foster and Williamson proved that G contains a spanning closed trail if |V(G)| = n ≥ 6, δ(G) ≥ 2 and deg u + deg v ≥ n – 1 for every pair of nonadjacent vertices u and v. The bound n – 1 can be decreased to (2n + 3)/3 if G is connected and bridgeless, which is necessary for G to have a spanning closed trail.     

The interval number of a planar graph: Three intervals suffice

Suppose each vertex of a graph G is assigned a subset of the real line consisting of at most t closed intervals. This assignment is called a t-interval representation of G when vertex v is adjacent to vertex w if and only if some interval for v intersects some interval for w. The interval number i(G) of a graph G is the smallest number t such that G has a t-interval representation. It is proved that i(G) ≤ 3 whenever G is planar and that this bound is the best possible. The related concepts of displayed interval number and depth-r interval number are discussed and their maximum values for certain classes of planar graphs are found.     

Closure Concepts: A Survey

In this paper we survey results of the following type (known as closure results). Let P be a graph property, and let C(u,v) be a condition on two nonadjacent vertices u and v of a graph G. Then G+uv has property P if and only if G has property P. The first and now well-known result of this type was established by Bondy and Chvátal in a paper published in 1976: If u and v are two nonadjacent vertices with degree sum n in a graph G on n vertices, then G+uv is hamiltonian if and only if G is hamiltonian. Based on this result, they defined the n-closure cl_n (G) of a graph G on n vertices as the graph obtained from G by recursively joining pairs of nonadjacent vertices with degree sum n until no such pair remains. They showed that cl_n(G) is well-defined, and that G is hamiltonian if and only if cl_n(G) is hamiltonian. Moreover, they showed that cl_n(G) can be obtained by a polynomial algorithm, and that a Hamilton cycle in cl_n(G) can be transformed into a Hamilton cycle of G by a polynomial algorithm. As a consequence, for any graph G with cl_n(G)=K _n (and n≥3), a Hamilton cycle can be found in polynomial time, whereas this problem is NP-hard for general graphs. All classic sufficient degree conditions for hamiltonicity imply a complete n-closure, so the closure result yields a common generalization as well as an easy proof for these conditions. In their first paper on closures, Bondy and Chvátal gave similar closure results based on degree sum conditions for nonadjacent vertices for other graph properties. Inspired by their first results, many authors developed other closure concepts for a variety of graph properties, or used closure techniques as a tool for obtaining deeper sufficiency results with respect to these properties. Our aim is to survey this progress on closures made in the past (more than) twenty years. Revised: September 27, 1999     

Minimal rankings and the arank number of a path

Given a graph G, a function f:V(G)→{1,2,…,k} is a k-ranking of G if f(u)=f(v) implies every u-v path contains a vertex w such that f(w)>f(u). A k-ranking is minimal if the reduction of any label greater than 1 violates the described ranking property. The arank number of a graph, denoted ψ_r(G), is the largest k such that G has a minimal k-ranking. We present new results involving minimal k-rankings of paths. In particular, we determine ψ_r(P_n), a problem posed by Laskar and Pillone in 2000.     

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     

On the Super Edge-Magic Deficiency of Graphs

A (p, q) graph G is edge-magic if there exists a bijective function f: V(G) ∪ E(G) → {1,2,…,p + q} such that f(u) + f(v) + f(uv) = k is a constant, called the valence of f, for any edge uv of G. Moreover, G is said to be super edge-magic if f(V(G)) = {1,2,…,p}. The question studied in this paper is for which graphs is it possible to add a finite number of isolated vertices so that the resulting graph is super edge-magic? If it is possible for a given graph G, then we say that the minimum such number of isolated vertices is the super edge-magic deficiency, μ_s(G) of G; otherwise we define it to be + ∞.     

Transformation graph

The transformation graph G^-+- of a graph G is the graph with vertex set V(G)E(G), in which two vertices u and v are joined by an edge if one of the following conditions holds: (i) u,vV(G) and they are not adjacent in G, (ii) u,vE(G) and they are adjacent in G, (iii) one of u and v is in V(G) while the other is in E(G), and they are not incident in G. In this paper, for any graph G, we determine the connectivity and the independence number of G^-+-. Furthermore, for a graph G of order n4, we show that G^-+- is hamiltonian if and only if G is not isomorphic to any graph in {2K₁+K₂,K₁+K₃}{K_1,n-1,K_1,n-1+e,K_1,n-2+K₁}.     

Edge disjoint Hamilton cycles in graphs

Let G be a graph of order n and k ≥ 0 an integer. It is conjectured in [8] that if for any two vertices u and v of a 2(k + 1)‐connected graph G,d _G (u,v) = 2 implies that max{d(u;G), d(v;G)} ≥ (n/2) + 2k, then G has k + 1 edge disjoint Hamilton cycles. This conjecture is true for k = 0, 1 (see cf. [3] and [8]). It will be proved in this paper that the conjecture is true for every integer k ≥ 0. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 8–20, 2000