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 .     

A note on Roman domination in graphs

Let G=(V,E) be a simple graph. A subset S⊆V is a dominating set of G, if for any vertex u∈V-S, there exists a vertex v∈S such that uv∈E. The domination number of G, γ(G), equals the minimum cardinality of a dominating set. A Roman dominating function on graph G=(V,E) is a function f:V→{0,1,2} satisfying the condition that every vertex v for which f(v)=0 is adjacent to at least one vertex u for which f(u)=2. The weight of a Roman dominating function is the value f(V)=∑_v∈Vf(v). The Roman domination number of a graph G, denoted by γ_R(G), equals the minimum weight of a Roman dominating function on G. In this paper, for any integer k(2?k?γ(G)), we give a characterization of graphs for which γ_R(G)=γ(G)+k, which settles an open problem in [E.J. Cockayne, P.M. Dreyer Jr, S.M. Hedetniemi et al. On Roman domination in graphs, Discrete Math. 278 (2004) 11-22].     

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 + ∞.     

Graham’s pebbling conjecture on product of thorn graphs of complete graphs

The pebbling number of a graph G, f(G), is the least n such that, no matter how n pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of pebbling moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. Let p₁,p₂,…,p_n be positive integers and G be such a graph, V(G)=n. The thorn graph of the graph G, with parameters p₁,p₂,…,p_n, is obtained by attaching p_i new vertices of degree 1 to the vertex u_i of the graph G, i=1,2,…,n. Graham conjectured that for any connected graphs G and H, f(G×H)≤f(G)f(H). We show that Graham’s conjecture holds true for a thorn graph of the complete graph with every by a graph with the two-pebbling property. As a corollary, Graham’s conjecture holds when G and H are the thorn graphs of the complete graphs with every .     

Subdivided trees are integral sum graphs

A graph G=(V,E) is an integral sum graph (ISG) if there exists a labeling S(G)⊂Z such that V=S(G) and for every pair of distinct vertices u,v∈V, uv is an edge if and only if u+v∈V. A vertex in a graph is called a fork if its degree is not 2. In 1998, Chen proved that every tree whose forks are at distance at least 4 from each other is an ISG. In 2004, He et al. reduced the distance to 3. In this paper we reduce the distance further to 2, i.e. we prove that every tree whose forks are at least distance 2 apart is an ISG.     

On the extension of vertex maps to graph homomorphisms

A reflexive graph is a simple undirected graph where a loop has been added at each vertex. If G and H are reflexive graphs and U⊆V(H), then a vertex map f:U→V(G) is called nonexpansive if for every two vertices x,y∈U, the distance between f(x) and f(y) in G is at most that between x and y in H. A reflexive graph G is said to have the extension property (EP) if for every reflexive graph H, every U⊆V(H) and every nonexpansive vertex map f:U→V(G), there is a graph homomorphism φ_f:H→G that agrees with f on U. Characterizations of EP-graphs are well known in the mathematics and computer science literature. In this article we determine when exactly, for a given “sink”-vertex s∈V(G), we can obtain such an extension φ_f;s that maps each vertex of H closest to the vertex s among all such existing homomorphisms φ_f. A reflexive graph G satisfying this is then said to have the sink extension property (SEP). We then characterize the reflexive graphs with the unique sink extension property (USEP), where each such sink extensions φ_f;s is unique.     

Super edge-magic strength of fire crackers, banana trees and unicyclic graphs

A graph G(V,E) is called super edge-magic if there exists a bijection f from V∪E to {1,2,3,…,|V|+|E|} such that f(u)+f(v)+f(uv)=c(f) is constant for any uv∈E and f(V)={1,2,3,…,|V|}. Such a bijection is called a super edge-magic labeling of G. The super edge-magic strength of a graph G is defined as the minimum of all c(f) where the minimum runs over all super edge-magic labelings of G and is denoted by sm(G). The super edge-magic strength of some families of graphs are obtained in this paper.     

Minus total domination in graphs

A three-valued function f: V → {−1, 0, 1} defined on the vertices of a graph G= (V, E) is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. That is, for every υ ∈ V, f(N(υ)) ⩾ 1, where N(υ) consists of every vertex adjacent to υ. The weight of an MTDF is f(V) = Σf(υ), over all vertices υ ∈ V. The minus total domination number of a graph G, denoted γ _t ⁻(G), equals the minimum weight of an MTDF of G. In this paper, we discuss some properties of minus total domination on a graph G and obtain a few lower bounds for γ _t ⁻(G).     

Connected subgraphs with small degree sums in 3‐connected planar graphs

It is well‐known that every planar graph has a vertex of degree at most five. Kotzig proved that every 3‐connected planar graph has an edge xy such that deg(x) + deg (y) ≤ 13. In this article, considering a similar problem for the case of three or more vertices that induce a connected subgraph, we show that, for a given positive integer t, every 3‐connected planar graph G with |V(G)| ≥ t has a connected subgraph H of order t such that Σ_x∈V(H) deg_G(x) ≤ 8t − 1. As a tool for proving this result, we consider decompositions of 3‐connected planar graphs into connected subgraphs of order at least t and at most 2t − 1. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 191–203, 1999     

On the Ramsey-Turán numbers of graphs and hypergraphs

Let t be an integer, f(n) a function, and H a graph. Define the t-Ramsey-Turán number of H, RT _t (n,H, f(n)), to be the maximum number of edges in an n-vertex, H-free graph G with α _t (G) ≤ f(n), where α _t (G) is the maximum number of vertices in a K _t -free induced subgraph of G. Erd?s, Hajnal, Simonovits, Sós and Szemerédi [6] posed several open questions about RT _t (n,K _s , o(n)), among them finding the minimum ? such that RT _t (n,K _t+? , o(n)) = Ω(n ²), where it is easy to see that RT _t (n,K _t+1, o(n)) = o(n ²). In this paper, we answer this question by proving that RT _t (n,K _t+2, o(n)) = Ω(n ²); our constructions also imply several results on the Ramsey-Turán numbers of hypergraphs.     

Some properties onf-edge covered critical graphs

LetG(V, E) be a simple graph, and letf be an integer function onV with 1 ≤f(v) ≤d(v) to each vertexv ∈V. An f-edge cover-coloring of a graphG is a coloring of edge setE such that each color appears at each vertexv ∈V at leastf(v) times. Thef-edge cover chromatic index ofG, denoted by χ′ _fc (G), is the maximum number of colors such that anf-edge cover-coloring ofG exists. Any simple graphG has anf-edge cover chromatic index equal to δ_f or δ _f - 1, where $\delta _f = \mathop {\min }\limits_{\upsilon \in V} \{ \left\lfloor {\frac{{d(v)}}{{f(v)}}} \right\rfloor \} $ . LetG be a connected and not complete graph with χ′ _fc (G)=δ _f-1, if for eachu, v ∈V and e =uv ?E, we have ÷ _fc (G + e) > ÷ _fc (G), thenG is called anf-edge covered critical graph. In this paper, some properties onf-edge covered critical graph are discussed. It is proved that ifG is anf-edge covered critical graph, then for eachu, v ∈V and e =uv ?E there existsw ∈ {u, v } withd(w) ≤ δ _f (f(w) + 1) - 2 such thatw is adjacent to at leastd(w) - δ _f + 1 vertices which are all δ _f -vertex inG.     

Cycle-magic graphs

A simple graph G=(V,E) admits a cycle-covering if every edge in E belongs at least to one subgraph of G isomorphic to a given cycle C. Then the graph G is C-magic if there exists a total labelling f:V∪E→{1,2,…,|V|+|E|} such that, for every subgraph H^′=(V^′,E^′) of G isomorphic to C, ∑_v∈V^′f(v)+∑_e∈E^′f(e) is constant. When f(V)={1,…,|V|}, then G is said to be C-supermagic.We study the cyclic-magic and cyclic-supermagic behavior of several classes of connected graphs. We give several families of C_r-magic graphs for each r?3. The results rely on a technique of partitioning sets of integers with special properties.     

On Group Chromatic Number of Graphs

Let G be a graph and A an Abelian group. Denote by F(G, A) the set of all functions from E(G) to A. Denote by D an orientation of E(G). For f ∈ F(G,A), an (A,f)-coloring of G under the orientation D is a function c : V(G)↦A such that for every directed edge uv from u to v, c(u)−c(v) ≠ f(uv). G is A-colorable under the orientation D if for any function f ∈ F(G, A), G has an (A, f)-coloring. It is known that A-colorability is independent of the choice of the orientation. The group chromatic number of a graph G is defined to be the least positive integer m for which G is A-colorable for any Abelian group A of order ≥m, and is denoted by χ_g(G). In this note we will prove the following results. (1) Let H₁ and H₂ be two subgraphs of G such that V(H₁)∩V(H₂)=∅ and V(H₁)∪V(H₂)=V(G). Then χ_g(G)≤min{max{χ_g(H₁), max_v∈V(H2)deg(v,G)+1},max{χ_g(H₂), max_u∈V(H1) deg (u, G) + 1}}. We also show that this bound is best possible. (2) If G is a simple graph without a K_3,3-minor, then χ_g(G)≤5.     

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.     

Connectivity measures in matched sum graphs

A matched sum graph G of two graphs G₁ and G₂ of the same order is obtained from the union of G₁ and G₂ and from joining each vertex of G₁ with one vertex of G₂ according to one bijection f between the vertices in V(G₁) and V(G₂). When G₁=G₂=H then f is just a permutation of V(H) and the corresponding matched sum graph is a permutation graph H^f. In this paper, we derive lower bounds for the connectivity, edge-connectivity, and different conditional connectivities in matched sum graphs, and present sufficient conditions which guarantee maximum values for these conditional connectivities.     

Lower negative decision number in a graph

Let G=(V,E) be a graph. A function f:V(G)→{?1,1} is called bad if ∑ _v∈N(v) f(v)≤1 for every v∈V(G). A bad function f of a graph G is maximal if there exists no bad function g such that g≠f and g(v)≥f(v) for every v∈V. The minimum of the values of ∑ _v∈V f(v), taken over all maximal bad functions f, is called the lower negative decision number and is denoted by β _D ^* (G). In this paper, we present sharp lower bounds on this number for regular graphs and nearly regular graphs, and we also characterize the graphs attaining those bounds.     

Disjoint Kr-minors in large graphs with given average degree

It is proved that there are functions f(r) and N(r,s) such that for every positive integer r, s, each graph G with average degree d(G)=2|E(G)|/|V(G)|≥f(r), and with at least N(r,s) vertices has a minor isomorphic to K_r,s or to the union of s disjoint copies of K_r.     

Vertex Coverings by Coloured Induced Graphs—Frames and Umbrellas

A graph G homogeneously embeds in a graph H if for every vertex x of G and every vertex y of H there is an induced copy of G in H with x at y. The graph G uniformly embeds in H if for every vertex y of H there is an induced copy of G in H containing y. For positive integer k, let f_k(G) (respectively, g_k(G)) be the minimum order of a graph H whose edges can be k-coloured such that for each colour, G homogeneously embeds (respectively, uniformly embeds) in the graph given by V(H) and the edges of that colour. We investigate the values f₂(G) and g₂(G) for special classes of G, in particular when G is a star or balanced complete bipartite graph. Then we investigate f_k(G) and g_k(G) when k ≥ 3 and G is a complete graph.     

Weak sense of direction labelings and graph embeddings

An edge-labeling λ for a directed graph G has a weak sense of direction (WSD) if there is a function f that satisfies the condition that for any node u and for any two label sequences α and α^′ generated by non-trivial walks on G starting at u, f(α)=f(α^′) if and only if the two walks end at the same node. The function f is referred to as a coding function of λ. The weak sense of direction number of G, WSD(G), is the smallest integer k so that G has a WSD-labeling that uses k labels. It is known that WSD(G)≥Δ⁺(G), where Δ⁺(G) is the maximum outdegree of G.Let us say that a function τ:V(G)→V(H) is an embedding from G onto H if τ demonstrates that G is isomorphic to a subgraph of H. We show that there are deep connections between WSD-labelings and graph embeddings. First, we prove that when f_H is the coding function that naturally accompanies a Cayley graph H and G has a node that can reach every other node in the graph, then G has a WSD-labeling that has f_H as a coding function if and only if G can be embedded onto H. Additionally, we show that the problem “Given G, does G have a WSD-labeling that uses a particular coding function f?” is NP-complete even when G and f are fairly simple.Second, when D is a distributive lattice, H(D) is its Hasse diagram and G(D) is its cover graph, then WSD(H(D))=Δ⁺(H(D))=d^∗, where d^∗ is the smallest integer d so that H(D) can be embedded onto the d-dimensional mesh. Along the way, we also prove that the isometric dimension of G(D) is its diameter, and the lattice dimension of G(D) is Δ⁺(H(D)). Our WSD-labelings are poset-based, making use of Birkhoff’s characterization of distributive lattices and Dilworth’s theorem for posets.     

Distances in orientations of graphs

We prove that there is a function h(k) such that every undirected graph G admits an orientation H with the following property: If an edge uv belongs to a cycle of length k in G, then uv or vu belongs to a directed cycle of length at most h(k) in H. Next, we show that every undirected bridgeless graph of radius r admits an orientation of radius at most r² + r, and this bound is best possible. We consider the same problem with radius replaced by diameter. Finilly, we show that the problem of deciding whether an undirected graph admits an orientation of diameter (resp. radius) 2 belongs to a class of problems called NP-hard.