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Isomorphic components of direct products of bipartite graphs

Richard Hammack (2006)

Discussiones Mathematicae Graph Theory

A standard result states the direct product of two connected bipartite graphs has exactly two components. Jha, Klavžar and Zmazek proved that if one of the factors admits an automorphism that interchanges partite sets, then the components are isomorphic. They conjectured the converse to be true. We prove the converse holds if the factors are square-free. Further, we present a matrix-theoretic conjecture that, if proved, would prove the general case of the converse; if refuted, it would produce a...

Isomorphic components of Kronecker product of bipartite graphs

Pranava K. Jha, Sandi Klavžar, Blaž Zmazek (1997)

Discussiones Mathematicae Graph Theory

Weichsel (Proc. Amer. Math. Soc. 13 (1962) 47-52) proved that the Kronecker product of two connected bipartite graphs consists of two connected components. A condition on the factor graphs is presented which ensures that such components are isomorphic. It is demonstrated that several familiar and easily constructible graphs are amenable to that condition. A partial converse is proved for the above condition and it is conjectured that the converse is true in general.

Isomorphic digraphs from powers modulo p

Guixin Deng, Pingzhi Yuan (2011)

Czechoslovak Mathematical Journal

Let p be a prime. We assign to each positive number k a digraph G p k whose set of vertices is { 1 , 2 , ... , p - 1 } and there exists a directed edge from a vertex a to a vertex b if a k ≡ b ( mod p ) . In this paper we obtain a necessary and sufficient condition for G p k 1 ≃ G p k 2 .

Isomorphisms and traversability of directed path graphs

Hajo Broersma, Xueliang Li (2002)

Discussiones Mathematicae Graph Theory

The concept of a line digraph is generalized to that of a directed path graph. The directed path graph Pₖ(D) of a digraph D is obtained by representing the directed paths on k vertices of D by vertices. Two vertices are joined by an arc whenever the corresponding directed paths in D form a directed path on k+1 vertices or form a directed cycle on k vertices in D. In this introductory paper several properties of P₃(D) are studied, in particular with respect to isomorphism and traversability. In our...

Isonemality and mononemality of woven fabrics

Bohdan Zelinka (1983)

Aplikace matematiky

The paper studies the diagrams of woven fabrics consisting of white and black squares as geometrical objects and described their symmetries. The concepts of isonemality and mononemality due to B. Grünbaum and G. C. Shephard are used. A conjecture of these authors is proved in a particular case.

Iterated arc graphs

Danny Rorabaugh, Claude Tardif, David Wehlau, Imed Zaguia (2018)

Commentationes Mathematicae Universitatis Carolinae

The arc graph δ ( G ) of a digraph G is the digraph with the set of arcs of G as vertex-set, where the arcs of δ ( G ) join consecutive arcs of G . In 1981, S. Poljak and V. Rödl characterized the chromatic number of δ ( G ) in terms of the chromatic number of G when G is symmetric (i.e., undirected). In contrast, directed graphs with equal chromatic numbers can have arc graphs with distinct chromatic numbers. Even though the arc graph of a symmetric graph is not symmetric, we show that the chromatic number of the...

Iterated neighborhood graphs

Martin Sonntag, Hanns-Martin Teichert (2012)

Discussiones Mathematicae Graph Theory

The neighborhood graph N(G) of a simple undirected graph G = (V,E) is the graph ( V , E N ) where E N = a,b | a ≠ b, x,a ∈ E and x,b ∈ E for some x ∈ V. It is well-known that the neighborhood graph N(G) is connected if and only if the graph G is connected and non-bipartite. We present some results concerning the k-iterated neighborhood graph N k ( G ) : = N ( N ( . . . N ( G ) ) ) of G. In particular we investigate conditions for G and k such that N k ( G ) becomes a complete graph.

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