An upper bound for the k-domination number of a graph

The k-domination number of a graph G, γk(G), is the least cardinality of a set U of verticies such that any other vertex is adjacent to at least k vertices of U. We prove that if each vertex has degree at least k, then γk(G) ≤ kp/(k + 1).     

Bounds on the connected domination number of a graph

A subset S$S$ of vertices in a graph G=(V,E)$G=(V,E)$ is a connected dominating set of G$G$ if every vertex of V?S$V?S$ is adjacent to a vertex in S$S$ and the subgraph induced by S$S$ is connected. The minimum cardinality of a connected dominating set of G$G$ is the connected domination number _γc(G)${\gamma }_c\left(G\right)$. The girth g(G)$g\left(G\right)$ is the length of a shortest cycle in G$G$. We show that if G$G$ is a connected graph that contains at least one cycle, then _γc(G)≥g(G)−2${\gamma }_c\left(G\right)\ge g\left(G\right)-2$, and we characterize the graphs obtaining equality in this bound. We also establish various upper bounds on the connected domination number of a graph, as well as Nordhaus–Gaddum type results.     

Bounds for the covering number of a graph

Upper and lower bounds for the covering number of a graph are obtained. It is shown, by probabilistic methods, that there exists a large class of graphs for which the upper bound obtained is essentially best possible.     

Bounds for the harmonious chromatic number of a graph

The upper bound for the harmonious chromatic number of a graph given by Zhikang Lu and by C. McDiarmid and Luo Xinhua, independently (Journal of Graph Theory, 1991, pp. 345–347 and 629–636) and the lower bound given by D. G. Beane, N. L. Biggs, and B. J. Wilson (Journal of Graph Theory, 1989, pp. 291–298) are improved.     

Bounds for the ramsey number of a disconnected graph versus any graph

Bounds are determined for the Ramsey number of the union of graphs versus a fixed graph H, based on the Ramsey number of the components versus H. For certain unions of graphs, the exact Ramsey number is determined. From these formulas, some new Ramsey numbers are indicated. In particular, if . Where k_i is the number of components of order i and t₁ (H) is the minimum order of a color class over all critical colorings of the vertices of H, then .     

Bounds on the number of isolates in sum graph labeling

A simple undirected graph H is called a sum graph if there is a labeling L of the vertices of H into distinct positive integers such that any two vertices u and v of H are adjacent if and only if there is a vertex w with label L(w)=L(u)+L(v). The sum number σ(G) of a graph G=(V,E) is the least integer r such that the graph H consisting of G and r isolated vertices is a sum graph. It is clear that σ(G)|E|. In this paper, we discuss general upper and lower bounds on the sum number. In particular, we prove that, over all graphs G=(V,E) with fixed |V|3 and |E|, the average of σ(G) is at least

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. In other words, for most graphs, σ(G)Ω(|E|).     

Bounds on the number of cycles of length three in a planar graph

LetG be ap-vertex planar graph having a representation in the plane with nontriangular facesF ₁,F ₂, …,F _r. Letf ₁,f ₂, …,f _r denote the lengths of the cycles bounding the facesF ₁,F ₂, …,F _r respectively. LetC ₃(G) be the number of cycles of length three inG. We give bounds onC ₃(G) in terms ofp,f ₁,f ₂, …,f _r. WhenG is 3-connected these bounds are bounds for the number of triangles in a polyhedron. We also show that all possible values ofC ₃(G) between the maximum and minimum value are actually achieved. This research was supported in part by the U.S.A.F. Office of Scientific Research, Systems Command, under Grant AFOSR-76-3017 and the National Science Foundation under Grant ENG79-09724.     

On the average crosscap number II: Bounds for a graph

The bounds are obtained for the average crosscap number. Let G be a graph which is not a tree. It is shown that the average crosscap number of G is not less thanβ(G)-1/2β(G)-1β(G) and not larger thanβ(G). Furthermore, we also describe the structure of the graphs which attain the bounds of the average crosscap number.     

On the average crosscap number Ⅱ: Bounds for a graph

The bounds are obtained for the average crosscap number. Let G be a graph which is not a tree. It is shown that the average crosscap number of G is not less than 2β(G)-1/2β(G)-1β(G)β(G) and not larger than/β(G). Furthermore, we also describe the structure of the graphs which attain the bounds of the average crosscap number.     

Bounds on the chromatic polynomial and on the number of acyclic orientations of a graph

An upper bound is given on the number of acyclic orientations of a graph, in terms of the spectrum of its Laplacian. It is shown that this improves upon the previously known bound, which depended on the degree sequence of the graph. Estimates on the new bound are provided.A lower bound on the number of acyclic orientations of a graph is given, with the help of the probabilistic method. This argument can take advantage of structural properties of the graph: it is shown how to obtain stronger bounds for small-degree graphs of girth at least five, than are possible for arbitrary graphs. A simpler proof of the known lower bound for arbitrary graphs is also obtained.Both the upper and lower bounds are shown to extend to the general problem of bounding the chromatic polynomial from above and below along the negative real axis.Partially supported by the NSF under grant CCR-9404113. Most of this research was done while the author was at the Massachusetts Institute of Technology, and was supported by the Defense Advanced Research Projects Agency under Contracts N00014-92-J-1799 and N00014-91-J-1698, the Air Force under Contract F49620-92-J-0125, and the Army under Contract DAAL-03-86-K-0171.Supported by an ONR graduate fellowship, grants NSF 8912586 CCR and AFOSR 89-0271, and an NSF postdoctoral fellowship.     

Bounds for the number of meeting edges in graph partitioning

Let G be a weighted hypergraph with edges of size at most 2. Bollobás and Scott conjectured that G admits a bipartition such that each vertex class meets edges of total weight at least (w ₁?Δ₁)/2+2w ₂/3, where wi is the total weight of edges of size i and Δ₁ is the maximum weight of an edge of size 1. In this paper, for positive integer weighted hypergraph G (i.e., multi-hypergraph), we show that there exists a bipartition of G such that each vertex class meets edges of total weight at least (w ₀?1)/6+(w ₁?Δ₁)/3+2w ₂/3, where w ₀ is the number of edges of size 1. This generalizes a result of Haslegrave. Based on this result, we show that every graph with m edges, except for K ₂ and K _1,3, admits a tripartition such that each vertex class meets at least [2m/5] edges, which establishes a special case of a more general conjecture of Bollobás and Scott.     

Bounds on graph spectra

The largest eigenvalue of the adjacency matrix of a graph has received considerable attention in the literature. Not nearly as much seems to be known about bounds on other eigenvalues of the spectrum. Several results are presented here toward that goal, first for the general class of simple graphs, then for triangle-free graphs and finally for the even more restricted class of bipartite graphs.     

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.     

Bounds of eigenvalues of a graph

LetG be a simple graph withn vertices. We denote by _i(G) thei-th largest eigenvalue ofG. In this paper, several results are presented concerning bounds on the eigenvalues ofG. In particular, it is shown that –1₂(G)(n–2)/2, and the left hand equality holds if and only ifG is a complete graph with at least two vertices; the right hand equality holds if and only ifn is even andG2K _n/2.     

A lower bound for the distance k-domination number of trees

A subset D of vertices of a graph G = (V, E) is a distance k-dominating set for G if the distance between every vertex of V ? D and D is at most k. The minimum size of a distance k-dominating set of G is called the distance k-domination number γ_k(G) of G. In this paper we prove that (2k + 1)γ_k(T) ≥ ¦V¦ + 2k ? kn₁ for each tree T = (V, E) with n₁ leafs, and we characterize the class of trees that satisfy the equality (2k + 1)γ_k(T) = ¦V¦ + 2k ? kn₁. Our results generalize those of Lemanska [4] for k = 1 and of Cyman, Lemanska and Raczek [1] for k = 2.     

Bounds on a graph's security number

Let G=(V,E) be a graph. A set S⊆V is a defensive alliance if |N[x]∩S|?|N[x]-S| for every x∈S. Thus, each vertex of a defensive alliance can, with the aid of its neighbors in S, be defended from attack by its neighbors outside of S. An entire set S is secure if any subset X⊆S, not just singletons, can be defended from an attack from outside of S, under an appropriate definition of what such a defense implies. The security number s(G) of G is the cardinality of a smallest secure set. Bounds on s(G) are presented.     

Bounds on the localization number

We consider the localization game played on graphs, wherein a set of cops attempt to determine the exact location of an invisible robber by exploiting distance probes. The corresponding optimization parameter for a graph G is called the localization number and is written as ζ(G). We settle a conjecture of Bosek et al by providing an upper bound on the chromatic number as a function of the localization number. In particular, we show that every graph with ζ(G) ≤ k has degeneracy less than 3^k and, consequently, satisfies χ(G) ≤ 3^ζ(G). We show further that this degeneracy bound is tight. We also prove that the localization number is at most 2 in outerplanar graphs, and we determine, up to an additive constant, the localization number of hypercubes.     

Bounds on the index of the signless Laplacian of a graph

Let G=(V,E) be a simple, undirected graph of order n and size m with vertex set V, edge set E, adjacency matrix A and vertex degrees Δ=d₁≥d₂≥?≥d_n=δ. The average degree of the neighbor of vertex v_i is . Let D be the diagonal matrix of degrees of G. Then L(G)=D(G)−A(G) is the Laplacian matrix of G and Q(G)=D(G)+A(G) the signless Laplacian matrix of G. Let μ₁(G) denote the index of L(G) and q₁(G) the index of Q(G). We survey upper bounds on μ₁(G) and q₁(G) given in terms of the d_i and m_i, as well as the numbers of common neighbors of pairs of vertices. It is well known that μ₁(G)≤q₁(G). We show that many but not all upper bounds on μ₁(G) are still valid for q₁(G).