Search

The analogue of Hadwiger's conjecture for the immersion order states that every graph G contains the complete graph K X(G) as an immersion. Like its minor-order counterpart it is open even for graphs with independence number 2. Let G and H be graphs with independence number at most 2, such that |V(H)| ≤ 4. We show that if G is H -free, then G satisfies the conjecture. [ABSTRACT FROM AUTHOR]

Let n and k be positive integers with n ≥ 2 k . Consider a circle C with n points 1 , … , n in clockwise order. The interlacing graph IG n , k is the graph with vertices corresponding to k -subsets of [ n ] that do not contain two adjacent points on C , and edges between k -subsets P and Q if they interlace : after removing the points in P from C , the points in Q are in different connected components. In this paper we prove that the circular chromatic number of IG n , k is equal to n / k , hence the chromatic number is ⌈ n / k ⌉ , and that its independence number is ( n − k − 1 k − 1 ) . [ABSTRACT FROM AUTHOR]

The Grundy domination number, γ gr (G) , of a graph G is the maximum length of a sequence (v 1 , v 2 , ... , v k) of vertices in G such that for every i ∈ { 2 , ... , k } , the closed neighborhood N [ v i ] contains a vertex that does not belong to any closed neighborhood N [ v j ] , where j < i. It is well known that the Grundy domination number of any graph G is greater than or equal to the upper domination number Γ (G) , which is in turn greater than or equal to the independence number α (G). In this paper, we initiate the study of the class of graphs G with Γ (G) = γ gr (G) and its subclass consisting of graphs G with α (G) = γ gr (G). We characterize the latter class of graphs among all twin-free connected graphs, provide a number of properties of these graphs, and prove that the hypercubes are members of this class. In addition, we give several necessary conditions for graphs G with Γ (G) = γ gr (G) and present large families of such graphs. [ABSTRACT FROM AUTHOR]

The power graph Γ G of a finite group G is the graph whose vertex set is G , two distinct elements being adjacent if one is a power of the other. In this paper, we give sharp lower and upper bounds for the independence number of Γ G and characterize the groups achieving the bounds. Moreover, we determine the independence number of Γ G if G is cyclic, dihedral or generalized quaternion. Finally, we classify all finite groups G whose power graphs have independence number 3 or n − 2 , where n is the order of G . [ABSTRACT FROM AUTHOR]

In a graph G = ( V , E ) (not necessarily be connected) an independent set S ⊆ V is said to be an outer connected independent set if ω ( G − S ) ≤ ω ( G ) , where ω ( G ) is the number of components in G . The maximum cardinality of an outer connected independent set is called the outer connected independence number and it is denoted by I o c ( G ) . This concept was introduced in [I. Sahul Hamid, R. Gnanaprahasam, M. Fatima Mary Outer Connected Independence in Graphs, Discrete Math, Algorithm, Appl Vol. 7, No. 3 1550039 (2015) [11 pages] DOI]. This paper examine the effect of I o c when G is modified by deleting an edge. [ABSTRACT FROM AUTHOR]

We continue the work of Gyárfás, Győri and Simonovits [Gyárfás, A., E. Győri and M. Simonovits, On 3 -uniform hypergraphs without linear cycles . Journal of Combinatorics 7 (2016), 205–216], who proved that if a 3-uniform hypergraph H with n vertices has no linear cycles, then its independence number α ≥ 2 n 5 . The hypergraph consisting of vertex disjoint copies of complete hypergraphs K 5 3 shows that equality can hold. They asked whether α can be improved if we exclude K 5 3 as a subhypergraph and whether such a hypergraph is 2-colorable. We answer these questions affirmatively. Namely, we prove that if a 3-uniform linear-cycle-free hypergraph H , doesn't contain K 5 3 as a subhypergraph, then it is 2-colorable. This result clearly implies that α ≥ ⌈ n 2 ⌉ . We show that this bound is sharp. Gyárfás, Győri and Simonovits also proved that a linear-cycle-free 3-uniform hypergraph contains a vertex of strong degree at most 2. In this context, we show that a linear-cycle-free 3-uniform hypergraph has a vertex of degree at most n − 2 when n ≥ 10 . [ABSTRACT FROM AUTHOR]

Let G be a graph on n vertices with independence number α . What is the largest k -connected subgraph that G must contain? We prove that if n is sufficiently large ( n ≥ α 2 k + 1 will do ) , then G contains a k -connected subgraph on at least n / α vertices. This is sharp, since G might be the disjoint union of α equally-sized cliques. For k ≥ 3 and α = 2 , 3 , we shall prove that the same result holds for n ≥ 4 ( k − 1 ) and n ≥ 27 ( k − 1 ) 4 sharp. [ABSTRACT FROM AUTHOR]

In this paper we study three families of graphs, one is the graphs of order n with connectivity κ ( G ) ≤ k and minimum degree δ ( G ) ≥ k . We show that among the graphs in this family, the maximum spectral radius is obtained uniquely at K k + ( K δ − k + 1 ∪ K n − δ − 1 ) . Another family of the graphs we study is the family of bipartite graphs with order n and connectivity k . We show that among the graphs in this family the maximum spectral radius is obtained at a graph modified from K ⌊ n / 2 ⌋ , n − 1 − ⌊ n / 2 ⌋ . The third family of graphs we have studied is the family of graphs with order n , connectivity k and independence number r . We determine the graphs in this family that have the maximum spectral radius. [ABSTRACT FROM AUTHOR]

In this work we obtain several tight bounds for the differential of a graph and we relate the differential of a graph with some known parameters of the graph. [ABSTRACT FROM AUTHOR]

Abstract: For the Erdős-Rényi random graph G n,p , we consider the order of a largest vertex subset that induces a subgraph with average degree at most t. For the case when both p and t are fixed, this value is asymptotically almost surely concentrated on at most two explicitly given points. This generalises a result on the independence number of random graphs. For both the upper and lower bounds, we rely on large deviations inequalities for the binomial distribution. [Copyright &y& Elsevier]

Abstract: A directed dominating set in a directed graph is a set of vertices of such that every vertex has an adjacent vertex in with directed to . The directed domination number of , denoted by , is the minimum cardinality of a directed dominating set in . The directed domination number of a graph , denoted , is the maximum directed domination number over all orientations of . The directed domination number of a complete graph was first studied by Erdős [P. Erdős On a problem in graph theory, Math. Gaz. 47 (1963) 220–222], albeit in a disguised form. In this paper we prove a Greedy Partition Lemma for directed domination in oriented graphs. Applying this lemma, we obtain bounds on the directed domination number. In particular, if denotes the independence number of a graph , we show that . [Copyright &y& Elsevier]

Independence number of the graph is an important parameter in graph theory. Let G = (V(G), E(G)) be a simple finite undirected graph. A sub-set S of V(G) is an independent set and any two vertices of S aren't adjacent. So the independence number a(G) is the maximum cardinality of an independent set in G. The independence number of path power graph, Flower Shark and its related graph, multi-cone graph is studied. Firstly, the independent sets of them are constructed, so their lower bounds of the independence number are obtained. Secondly, it is proved that these bounds are also the upper bounds of those three graphs. Then, their exact values of the independence number are given. [ABSTRACT FROM AUTHOR]

Abstract: Let denote the minimum cardinality of a subset H in such that every word in this space differs in at most R coordinates from a multiple of a vector in H, where q is a prime or a prime power. In order to explore the symmetries of such coverings, we investigate a few algebraic properties of invariant sets under permutation. Extremal problems arising from invariant sets are also studied on a graph theoretical viewpoint. As an application, a new class of upper bounds on is constructed. [Copyright &y& Elsevier]

Abstract: An incidence graph of a given graph G, denoted by I(G), has its vertex set such that the pair of vertices is an edge of I(G) if and only if there exists at least one case of , , or . Study of Incidence graphs was made by Zhagn Zhong-fu et al. in [Zhagn Zhong-fu, Yao Bing, Li Jing-wen, Linu Lin-zhong, Wang Jian-fang, Xu Bao-gen, On Incidence Graphs, ARS combinatoria 87 (2008), 213-223]. The origin of Incidence graphs can be traced to a paper titled Incidence and Strong edge colorings of graphs by R.A. Brualdi, et al. [Richard A. Brualdi and Jennifer J. Quinn Massey, Incidence and strong edge colorings of graphs, Discrete Mathematics 122 (1993) 51-58]. In this paper, we make a study of domination and related parameters in Incidence graphs. [Copyright &y& Elsevier]

An edge-colored connected graph G is properly connected if between every pair of distinct vertices, there exists a path such that no two adjacent edges have the same color. Fujita (2019) introduced the optimal proper connection number pc opt (G) for a monochromatic connected graph G , to make a connected graph properly connected efficiently. More precisely, pc opt (G) is the smallest integer p + q when one converts a given monochromatic graph G into a properly connected graph by recoloring p edges with q colors. In this paper, we show that pc opt (G) has an upper bound in terms of the independence number α (G). Namely, we prove that for a connected graph G , pc opt (G) ≤ 5 α (G) − 1 2 . Moreover, for the case α (G) ≤ 3 , we improve the upper bound to 4, which is tight. [ABSTRACT FROM AUTHOR]

Abstract: For integers and , distributions of n points in the d-dimensional unit cube are investigated, such that the minimum volume of the convex hull determined by j of n points is large. Lower and upper bounds on these minimum volumes are given. For obtaining lower bounds, results on the independence number of non-uniform, linear hypergraphs are used, which might be of interest by their own. [Copyright &y& Elsevier]

Abstract: An offensive alliance in a graph is a set of vertices where for every vertex v in its boundary it holds that the majority of vertices in v''s closed neighborhood are in S. In the case of strong offensive alliance, strict majority is required. An alliance S is called global if it affects every vertex in V\S, that is, S is a dominating set of Γ. The global offensive alliance number (respectively, global strong offensive alliance number ) is the minimum cardinality of a global offensive (respectively, global strong offensive) alliance in Γ. In this paper we obtain several tight bounds on and in terms of several parameters of Γ. [Copyright &y& Elsevier]