An implicit degree ore-condition for pancyclicity of graphs

In 1989, Zhu, Li and Deng introduced the definition of implicit degree of a vertex v in a graph G, denoted by id(v). In this paper, we prove that if G is a 2-connected graph of order n such that id(u) + id(v) ≥ n for each pair of nonadjacent vertices u and v in G, then G is pancyclic unless G is bipartite, or else n = 4r, r ≥ 2 and G is isomorphic to F4r .     

Minimum implicit degree condition restricted to claws for hamiltonian cycles

Let id(v) denote the implicit degree of a vertex v in a graph G. We define G to be implicit 1-heavy (implicit 2-heavy) if at least one (two) of the end vertices of each induced claw has (have) implicit degree at least n/2. In this paper, we prove that: (a) Let G be a 2-connected graph of order n ≥ 3. If G is implicit 2-heavy and |N(u) ∩ N(v)| ≥ 2 for every pair of vertices u and v with d(u, v) = 2 and max{id(u), id(v)} < n/2, then G is hamiltonian. (b) Let G be a 3-connected graph of order n ≥ 3. If G is implicit 1-heavy and |N(u) ∩ N(v)| ≥ 2 for each pair of vertices u and v with d(u, v) = 2 and max{id(u), id(v)} < n/2, then G is hamiltonian.     

An implicit degree condition for relative length of long paths and cycles in graphs

For a graph G, we denote by p(G) and c(G) the number of vertices of a longest path and a longest cycle in G, respectively. For a vertex v in G, id(v) denotes the implicit degree of v. In this paper, we obtain that if G is a 2-connected graph on n vertices such that the implicit degree sum of any three independent vertices is at least n + 1, then either G contains a hamiltonian path, or c(G) ≥ p(G) ? 1.     

Decomposition of complete multigraphs into crown graphs

Let λ K _v be the complete multigraph, G a finite simple graph. A G-design of λ K _v is denoted by GD(v,G,λ). The crown graph Q _n is obtained by joining single pendant edge to each vertex of an n-cycle. We give new constructions for Q _n -designs. Let v and λ be two positive integers. For n=4, 6, 8 and λ≥1, there exists a GD(v,Q _n ,λ) if and only if either (1) v>2n and λ v(v?1)≡0 (mod 4n), or (2) v=2n and λ≡0 (mod 4). Let n≥4 be even. Then (1) there exists a GD(2n,Q _n ,λ) if and only if λ≡0 (mod 4). (2) There exists a GD(2n+1,Q _n ,λ) when λ≡0 (mod 4).     

Hamilton Cycles in Implicit 2-Heavy Graphs

Let id(v) denote the implicit degree of a vertex v in a graph G. We define G of order n to be implicit 2-heavy if at least two of the end vertices of each induced claw have implicit degree at least \(\frac{n}{2}\). In this paper, we show that every implicit 2-heavy graph G is hamiltonian if we impose certain additional conditions on the connectivity of G or forbidden induced subgraphs. Our results extend two previous theorems of Broersma et al. (Discret Math 167–168:155–166, 1997) on the existence of Hamilton cycles in 2-heavy graphs.     

The Connected Monophonic Number of a Graph

For a connected graph G = (V, E) of order at least two, a chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called a monophonic path if it is a chordless path. A set S of vertices of G is a monophonic set of G if each vertex v of G lies on an x ? y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is defined as the monophonic number of G, denoted by m(G). A connected monophonic set of G is a monophonic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected monophonic set of G is the connected monophonic number of G and is denoted by m _c (G). We determine bounds for it and characterize graphs which realize these bounds. For any two vertices u and v in G, the monophonic distance d _m (u, v) from u to v is defined as the length of a longest u ? v monophonic path in G. The monophonic eccentricity e _m (v) of a vertex v in G is the maximum monophonic distance from v to a vertex of G. The monophonic radius rad _m G of G is the minimum monophonic eccentricity among the vertices of G, while the monophonic diameter diam _m G of G is the maximum monophonic eccentricity among the vertices of G. It is shown that for positive integers r, d and n ≥ 5 with r <  d, there exists a connected graph G with rad _m G =  r, diam _m G =  d and m _c (G) =  n. Also, if a,b and p are positive integers such that 2 ≤  a <  b ≤  p, then there exists a connected graph G of order p, m(G) =  a and m _c (G) =  b.     

A note on graphs whose neighborhoods are n-cycles

Let G be a graph, and let v be a vertex of G. We denote by N(v) the set of vertices of G which are adjacent to v, and by 〈N(v)〉 the subgraph of G induced by N(v). We call 〈N(v)〉 the neighborhood of v. In a paper of 1968, Agakishieva has, as one of her main theorems, the statement: “Graphs in which every neighborhood is an n-cycle exist if and only if 3?n?6.” It it is the object of this note to provide a list of counter examples to this statement.     

On the Evaluation of the Tutte Polynomial at the Points (1, –1) and (2, –1)

Motivated by the identity t (K _n+2; 1, –1) = t (K _n ; 2, –1), where t(G; x, y) is the Tutte polynomial of a graph G, we search for graphs G having the property that there is a pair of vertices u, v such that t(G; 1, –1) = t(G – {u, v}; 2, –1). Our main result gives a sufficient condition for a graph to have this property; moreover, it describes the graphs for which the property still holds when each vertex is replaced by a clique or a coclique of arbitrary order. In particular, we show that the property holds for the class of threshold graphs, a well-studied class of perfect graphs.     

On the rank spread of graphs

For a simple graph G?=?(𝒱, ?) with vertex-set 𝒱?=?{1,?…?,?n}, let 𝒮(G) be the set of all real symmetric n-by-n matrices whose graph is G. We present terminology linking established as well as new results related to the minimum rank problem, with spectral properties in graph theory. The minimum rank mr(G) of G is the smallest possible rank over all matrices in 𝒮(G). The rank spread r _v (G) of G at a vertex v, defined as mr(G)???mr(G???v), can take values ??∈?{0,?1,?2}. In general, distinct vertices in a graph may assume any of the three values. For ??=?0 or 1, there exist graphs with uniform r _v (G) (equal to the same integer at each vertex v). We show that only for ??=?0, will a single matrix A in 𝒮(G) determine when a graph has uniform rank spread. Moreover, a graph G, with vertices of rank spread zero or one only, is a λ-core graph for a λ-optimal matrix A in 𝒮(G). We also develop sufficient conditions for a vertex of rank spread zero or two and a necessary condition for a vertex of rank spread two.     

Sphere coverings and identifying codes

In any connected, undirected graph G = (V, E), the distance d(x, y) between two vertices x and y of G is the minimum number of edges in a path linking x to y in G. A sphere in G is a set of the form S _r (x) = {y ∈ V : d(x, y) = r}, where x is a vertex and r is a nonnegative integer called the radius of the sphere. We first address in this paper the following question: What is the minimum number of spheres with fixed radius r ≥ 0 required to cover all the vertices of a finite, connected, undirected graph G? We then turn our attention to the Hamming Hypercube of dimension n, and we show that the minimum number of spheres with any radii required to cover this graph is either n or n + 1, depending on the parity of n. We also relate the two above problems to other questions in combinatorics, in particular to identifying codes.     

Bounds on the 2-Rainbow Domination Number of Graphs

A 2-rainbow domination function of a graph G is a function f that assigns to each vertex a set of colors chosen from the set {1, 2}, such that for any ${v\in V(G), f(v)=\emptyset}$ implies ${\bigcup_{u\in N(v)}f(u)=\{1,2\}.}$ The 2-rainbow domination number γ _r2(G) of a graph G is the minimum ${w(f)=\Sigma_{v\in V}|f(v)|}$ over all such functions f. Let G be a connected graph of order |V(G)| = n ≥ 3. We prove that γ _r2(G) ≤ 3n/4 and we characterize the graphs achieving equality. We also prove a lower bound for 2-rainbow domination number of a tree using its domination number. Some other lower and upper bounds of γ _r2(G) in terms of diameter are also given.     

Lower bounds on the signed total k-domination number of graphs

Let G be a graph with vertex set V(G). For any integer k ≥ 1, a signed total k-dominating function is a function f: V(G) → {?1, 1} satisfying ∑_x∈N(v)f(x) ≥ k for every v ∈ V(G), where N(v) is the neighborhood of v. The minimum of the values ∑_v∈V(G)f(v), taken over all signed total k-dominating functions f, is called the signed total k-domination number. In this note we present some new sharp lower bounds on the signed total k-domination number of a graph. Some of our results improve known bounds.     

Rainbow Connection Number and Radius

The rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of its vertices is connected by at least one path in which no two edges are coloured the same. In this note we show that for every bridgeless graph G with radius r, rc(G) ≤  r(r + 2). We demonstrate that this bound is the best possible for rc(G) as a function of r, not just for bridgeless graphs, but also for graphs of any stronger connectivity. It may be noted that for a general 1-connected graph G, rc(G) can be arbitrarily larger than its radius (K _1,n for instance). We further show that for every bridgeless graph G with radius r and chordality (size of a largest induced cycle) k, rc(G) ≤  rk. Hitherto, the only reported upper bound on the rainbow connection number of bridgeless graphs is 4n/5 ? 1, where n is order of the graph (Caro et al. in Electron J Comb 15(1):Research paper 57, 13, 2008). It is known that computing rc(G) is NP-Hard (Chakraborty and fischer in J Comb Optim 1–18, 2009). Here, we present a (r + 3)-factor approximation algorithm which runs in O(nm) time and a (d + 3)-factor approximation algorithm which runs in O(dm) time to rainbow colour any connected graph G on n vertices, with m edges, diameter d and radius r.     

Large dense neighbourhoods and turán's theorem

Let t_r(n) denote the number of edges in the complete r-partite graph on n vertices whose colour classes are as equal in size as possible. It is proved that if G is a simple graph on n vertices and more than t_r(n) edges, and if v is any vertex of maximum degree m in G, then the subgraph of G induced by the neighbours of v has more than t_r?1(m) edges. Examples are given to demonstrate the sharpness of this theorem, and its algorithmic implications are noted.     

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     

Matching Properties in Total Domination Vertex Critical Graphs

A vertex subset S of a graph G = (V,E) is a total dominating set if every vertex of G is adjacent to some vertex in S. The total domination number of G, denoted by γ _t (G), is the minimum cardinality of a total dominating set of G. A graph G with no isolated vertex is said to be total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, γ _t (G?v) < γ _t (G). A total domination vertex critical graph G is called k-γ _t -critical if γ _t (G) = k. In this paper we first show that every 3-γ _t -critical graph G of even order has a perfect matching if it is K _1,5-free. Secondly, we show that every 3-γ _t -critical graph G of odd order is factor-critical if it is K _1,5-free. Finally, we show that G has a perfect matching if G is a K _1,4-free 4-γ _t (G)-critical graph of even order and G is factor-critical if G is a K _1,4-free 4-γ _t (G)-critical graph of odd order.     

Neighborhood contraction in graphs

Let G be a graph and v be any vertex of G. Then the neighborhood contracted graphG_v of G, with respect to the vertex v, is the graph with vertex set V ? N(v), where two vertices u,w ∈ V ? N(v) are adjacent in G_v if either w = v and u is adjacent to any vertex of N(v) in G or u,w ? N[v] and u,w are adjacent in G. The properties of the neighborhood contracted graphs are discussed in this paper. The neighborhood contraction in some special class of graphs, the domination in a graph and the neighborhood contracted graphs are discussed in the paper.     

Constructing and classifying neighborhood anti-Sperner graphs

For a simple graph G let N_G(u) be the (open) neighborhood of vertex u∈V(G). Then G is neighborhood anti-Sperner (NAS) if for every u there is a v∈V(G)?{u} with N_G(u)⊆N_G(v). And a graph H is neighborhood distinct (ND) if every neighborhood is distinct, i.e., if N_H(u)≠N_H(v) when u≠v, for all u and v∈V(H).In Porter and Yucas [T.D. Porter, J.L. Yucas. Graphs whose vertex-neighborhoods are anti-sperner, Bulletin of the Institute of Combinatorics and its Applications 44 (2005) 69-77] a characterization of regular NAS graphs was given: ‘each regular NAS graph can be obtained from a host graph by replacing vertices by null graphs of appropriate sizes, and then joining these null graphs in a prescribed manner’. We extend this characterization to all NAS graphs, and give algorithms to construct all NAS graphs from host ND graphs. Then we find and classify all connected r-regular NAS graphs for r=0,1,…,6.     

On Eccentric Connectivity Index of Eccentric Graph of Regular Dendrimer

The eccentric connectivity index \(\xi ^c(G)\) of a connected graph G is defined as \(\xi ^c(G) =\sum _{v \in V(G)}{deg(v) e(v)},\) where deg(v) is the degree of vertex v and e(v) is the eccentricity of v. The eccentric graph, \(G_e\), of a graph G has the same set of vertices as G,  with two vertices u, v adjacent in \(G_e\) if and only if either u is an eccentric vertex of v or v is an eccentric vertex of u. In this paper, we obtain a formula for the eccentric connectivity index of the eccentric graph of a regular dendrimer. We also derive a formula for the eccentric connectivity index for the second iteration of eccentric graph of regular dendrimer.     

On the maximal eccentric connectivity indices of graphs

For a connected simple graph G, the eccentricity ec(v) of a vertex v in G is the distance from v to a vertex farthest from v, and d(v) denotes the degree of a vertex v. The eccentric connectivity index of G, denoted by ξc(G), is defined as v∈V(G)d(v)ec(v). In this paper, we will determine the graphs with maximal eccentric connectivity index among the connected graphs with n vertices and m edges(n ≤ m ≤ n + 4), and propose a conjecture on the graphs with maximal eccentric connectivity index among the connected graphs with n vertices and m edges(m ≥ n + 5).