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).     

An upper bound for the harmonious chromatic number of a graph

An upper bound for the harmonious chromatic number of a graph G is given. Three corollaries of the theorem are theorems or improvements of the theorems of Miller and Pritikin.     

An upper bound on the path number of a digraph

A proof is presented of the conjecture of Alspach and Pullman that for any digraph G on n ≥ 4 vertices, the path number of G is at most [$\text{n}{}^2\text{4}$].     

An upper bound for the number of Eulerian orientations of a regular graph

We establish a new upper bound for the number of Eulerian orientations of a regular graph with even degrees.C.N.R.S., Paris with partial support of P.R.C. Mathématiques-Informatique.     

An upper bound for the adjacent vertex distinguishing acyclic edge chromatic number of a graph

A proper k-edge coloring of a graph G is called adjacent vertex distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the color set of edges incident to u is not equal to the color set of edges incident to υ, where uυ ∈ E(G). The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by χ′ _aa (G), is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. In this paper we prove that if G(V, E) is a graph with no isolated edges, then χ′ _aa (G) ≤ 32Δ. Supported by the Natural Science Foundation of Gansu Province (3ZS051-A25-025)     

On an upper bound for the harmonious chromatic number of a graph

The upper bound for the harmonious chromatic number of a graph that has been given by Sin-Min Lee and John Mitchem is improved.     

An upper bound on the domination number of a graph with minimum degree 2

A set S of vertices in a graph G is a dominating set of G if every vertex of V(G)?S is adjacent to some vertex in S. The minimum cardinality of a dominating set of G is the domination number of G, denoted as γ(G). Let P_n and C_n denote a path and a cycle, respectively, on n vertices. Let k₁(F) and k₂(F) denote the number of components of a graph F that are isomorphic to a graph in the family {P₃,P₄,P₅,C₅} and {P₁,P₂}, respectively. Let L be the set of vertices of G of degree more than 2, and let G−L be the graph obtained from G by deleting the vertices in L and all edges incident with L. McCuaig and Shepherd [W. McCuaig, B. Shepherd, Domination in graphs with minimum degree two, J. Graph Theory 13 (1989) 749-762] showed that if G is a connected graph of order n≥8 with δ(G)≥2, then γ(G)≤2n/5, while Reed [B.A. Reed, Paths, stars and the number three, Combin. Probab. Comput. 5 (1996) 277-295] showed that if G is a graph of order n with δ(G)≥3, then γ(G)≤3n/8. As an application of Reed’s result, we show that if G is a graph of order n≥14 with δ(G)≥2, then .     

An upper bound for the minimum rank of a graph

For a graph G of order n, the minimum rank of G is defined to be the smallest possible rank over all real symmetric n×n matrices A whose (i,j)th entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. We prove an upper bound for minimum rank in terms of minimum degree of a vertex is valid for many graphs, including all bipartite graphs, and conjecture this bound is true over for all graphs, and prove a related bound for all zero-nonzero patterns of (not necessarily symmetric) matrices. Most of the results are valid for matrices over any infinite field, but need not be true for matrices over finite fields.     

An upper bound on the paired-domination number in terms of the number of edges in the graph

In this paper, we continue the study of paired domination in graphs introduced by Haynes and Slater [T.W. Haynes, P.J. Slater, Paired-domination in graphs, Networks 32 (1998) 199-206]. A paired-dominating set of a graph is a dominating set whose induced subgraph contains a perfect matching. The paired-domination number of a graph G, denoted by , is the minimum cardinality of a paired-dominating set in G. We show that if G is a connected graph of size m≥18 with minimum degree at least 2, then and we characterize the (infinite family of) graphs that achieve equality in this bound.     

An upper bound for the number of matroids

An upper bound for the number of matroids is obtained. This upper bound complements the lower bound obtained by Piff and Welsh in [1].     

An upper bound for the total chromatic number

The total chromatic number,(G), of a graphG, is defined to be the minimum number of colours needed to colour the vertices and edges of a graph in such a way that no adjacent vertices, no adjacent edges and no incident vertex and edge are given the same colour. This paper shows that , where(G) is the vertex chromatic number and(G) is the edge chromatic number of the graph.Partially supported by ORS grant ORS/84120     

An upper bound for the restricted online Ramsey number

The restricted $(m,n;N)$-online Ramsey game is a game played between two players, Builder and Painter. The game starts with $N$ isolated vertices. Each turn Builder picks an edge to build and Painter chooses whether that edge is red or blue, and Builder aims to create a red $K_m$ or blue $K_n$ in as few turns as possible. The restricted online Ramsey number $\stackrel{?}{r}(m,n;N)$ is the minimum number of turns that Builder needs to guarantee her win in the restricted $(m,n;N)$-online Ramsey game. We show that if $N=r(n,n)$, $\stackrel{?}{r}(n,n;N)\le \left(\genfrac{}{}{0ex}{}{N}{2}\right)?\Omega (NlogN),$motivated by a question posed by Conlon, Fox, Grinshpun and He. The equivalent game played on infinitely many vertices is called the online Ramsey game. As almost all known Builder strategies in the online Ramsey game end up reducing to the restricted setting, we expect further progress on the restricted online Ramsey game to have applications in the general case.     

An improved edge bound on the interval number of a graph

The upper bound on the interval number of a graph in terms of its number of edges is improved. Also, the interval number of graphs in hereditary classes is bounded in terms of the vertex degrees.     

An upper bound on the number of planarK-sets

Given a setS ofn points, a subsetX of sizek is called ak-set if there is a hyperplane that separatesX fromS–X. We prove thatO(nk/log*k) is an upper bound for the number ofk-sets in the plane, thus improving the previous bound of Erdös, Lovász, Simmons, and Strauss by a factor of log*k.The research of J. Pach was supported in part by NSF Grant CCR-8901484 and by Grant OTKA-1418 from the Hungarian Foundation for Scientific Research. The research of W. Steiger and E. Szemerédi was supported in part by NSF Grant CCR-8902522. All authors express gratitude to the NSF DIMACS Center at Rutgers.     

An upper bound for the chromatic number of line graphs

It was conjectured by Reed [B. Reed, ω,α, and χ, Journal of Graph Theory 27 (1998) 177–212] that for any graph G, the graph’s chromatic number χ(G) is bounded above by , where Δ(G) and ω(G) are the maximum degree and clique number of G, respectively. In this paper we prove that this bound holds if G is the line graph of a multigraph. The proof yields a polynomial time algorithm that takes a line graph G and produces a colouring that achieves our bound.     

A lower bound for the interval number of a graph

The interval number i(G) of a graph G with n vertices is the lowest integer m such that G is the intersection graph of some family of sets I₁,…,I_n with every I_i being the union of at most m real intervals. In this article a lower bound for i(G) is proved followed by some considerations about the construction of graphs that are critical with respect to the interval number.     

An upper bound on the size of a largest clique in a graph

A graph is point determining if distinct vertices have distinct neighborhoods. The nucleus of a point-determining graph is the set G^O of all vertices, v, such that G–v is point determining. In this paper we show that the size, ω(G), of a maximum clique in G satisfies ω(G) ? 2|π (G)^O|, where π(G) (the point determinant of G) is obtained from G by identifying vertices which have the same neighborhood.