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Some results on the co-intersection graph of submodules of a module

Lotf Ali Mahdavi, Yahya Talebi (2018)

Commentationes Mathematicae Universitatis Carolinae

Let R be a ring with identity and M be a unitary left R -module. The co-intersection graph of proper submodules of M , denoted by Ω ( M ) , is an undirected simple graph whose vertex set V ( Ω ) is a set of all nontrivial submodules of M and two distinct vertices N and K are adjacent if and only if N + K ≠ M . We study the connectivity, the core and the clique number of Ω ( M ) . Also, we provide some conditions on the module M , under which the clique number of Ω ( M ) is infinite and Ω ( M ) is a planar graph. Moreover, we give several...

Some results on the recognizability of the linear groups over the binary field

Mohammad Reza Darafsheh, Yaghoub Farjami, M. Khademi, Ali Reza Moghaddamfar (2005)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we first find the set of orders of all elements in some special linear groups over the binary field. Then, we will prove the characterizability of the special linear group PSL ( 13 , 2 ) using only the set of its element orders.

Some results on total domination in direct products of graphs

Paul Dorbec, Sylvain Gravier, Sandi Klavžar, Simon Spacapan (2006)

Discussiones Mathematicae Graph Theory

Upper and lower bounds on the total domination number of the direct product of graphs are given. The bounds involve the {2}-total domination number, the total 2-tuple domination number, and the open packing number of the factors. Using these relationships one exact total domination number is obtained. An infinite family of graphs is constructed showing that the bounds are best possible. The domination number of direct products of graphs is also bounded from below.

Some Sharp Bounds on the Negative Decision Number of Graphs

Hongyu Liang (2013)

Discussiones Mathematicae Graph Theory

Let G = (V,E) be a graph. A function f : V → {-1,1} is called a bad function of G if ∑u∈NG(v) f(u) ≤ 1 for all v ∈ V where NG(v) denotes the set of neighbors of v in G. The negative decision number of G, introduced in [12], is the maximum value of ∑v∈V f(v) taken over all bad functions of G. In this paper, we present sharp upper bounds on the negative decision number of a graph in terms of its order, minimum degree, and maximum degree. We also establish a sharp Nordhaus-Gaddum-type inequality for...

Some sufficient conditions on odd directed cycles of bounded length for the existence of a kernel

Hortensia Galeana-Sánchez (2004)

Discussiones Mathematicae Graph Theory

A kernel N of a digraph D is an independent set of vertices of D such that for every w ∈ V(D)-N there exists an arc from w to N. If every induced subdigraph of D has a kernel, D is said to be a kernel-perfect digraph. In this paper I investigate some sufficient conditions for a digraph to have a kernel by asking for the existence of certain diagonals or symmetrical arcs in each odd directed cycle whose length is at most 2α(D)+1, where α(D) is the maximum cardinality of an independent vertex set...

Some totally 4-choosable multigraphs

Douglas R. Woodall (2007)

Discussiones Mathematicae Graph Theory

It is proved that if G is multigraph with maximum degree 3, and every submultigraph of G has average degree at most 2(1/2) and is different from one forbidden configuration C⁺₄ with average degree exactly 2(1/2), then G is totally 4-choosable; that is, if every element (vertex or edge) of G is assigned a list of 4 colours, then every element can be coloured with a colour from its own list in such a way that no two adjacent or incident elements are coloured with the same colour. This shows that the...

Some totally modular cordial graphs

Ibrahim Cahit (2002)

Discussiones Mathematicae Graph Theory

In this paper we define total magic cordial (TMC) and total sequential cordial (TSC) labellings which are weaker versions of magic and simply sequential labellings of graphs. Based on these definitions we have given several results on TMC and TSC graphs.

Some Toughness Results in Independent Domination Critical Graphs

Nawarat Ananchuen, Watcharaphong Ananchuen (2015)

Discussiones Mathematicae Graph Theory

A subset S of V (G) is an independent dominating set of G if S is independent and each vertex of G is either in S or adjacent to some vertex of S. Let i(G) denote the minimum cardinality of an independent dominating set of G. A graph G is k-i-critical if i(G) = k, but i(G+uv) < k for any pair of non-adjacent vertices u and v of G. In this paper, we establish that if G is a connected 3-i-critical graph and S is a vertex cutset of G with |S| ≥ 3, then [...] improving a result proved by Ao [3],...

Some Variations of Perfect Graphs

Magda Dettlaff, Magdalena Lemańska, Gabriel Semanišin, Rita Zuazua (2016)

Discussiones Mathematicae Graph Theory

We consider (ψk−γk−1)-perfect graphs, i.e., graphs G for which ψk(H) = γk−1(H) for any induced subgraph H of G, where ψk and γk−1 are the k-path vertex cover number and the distance (k − 1)-domination number, respectively. We study (ψk−γk−1)-perfect paths, cycles and complete graphs for k ≥ 2. Moreover, we provide a complete characterisation of (ψ2 − γ1)- perfect graphs describing the set of its forbidden induced subgraphs and providing the explicit characterisation of the structure of graphs belonging...

Spanning caterpillars with bounded diameter

Ralph Faudree, Ronald Gould, Michael Jacobson, Linda Lesniak (1995)

Discussiones Mathematicae Graph Theory

A caterpillar is a tree with the property that the vertices of degree at least 2 induce a path. We show that for every graph G of order n, either G or G̅ has a spanning caterpillar of diameter at most 2 log n. Furthermore, we show that if G is a graph of diameter 2 (diameter 3), then G contains a spanning caterpillar of diameter at most c n 3 / 4 (at most n).

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