A graph G is said to be hom-idempotent if there is a homomorphism from G2 to G, and weakly hom-idempotent if for some n ≥ 1 there is a homomorphism from Gn+1 to Gn . We characterize both classes of graphs in terms of a special class of Cayley graphs called normal Cayley graphs. This allows us to construct, for any integer n, a Cayley graph G such that Gn+1 → Gn 6→ Gn−1, answering a question of ...

Recibido: 10-10-2021.Aprobado: 15-11-2021.
 La historia de Vieja Habana prerrevolucionaria ofrece todo un reto para su estudio, tanto hacerlo en el país Cuba como a la distancia. Con ese mente, Ana Vera Estrada entrevistó, durante los últimos meses del año 1999 y primeros 2000, una serie adultos mayores, que, desde recuerdo memoria, dibujaran espacio llamado Habana: sus límites espaciales,...

A Estrada de Ferro Vitória a Minas (EFVM) sofreu, ao longo mais um século uso ininterrupto, diversas modificações em sua forma espacial e suas funções. Essas passaram despercebidas pelos diversos autores que dedicaram ela alguma atenção, lacuna o presente trabalho visa resolver apresentando novos mapas dados. Além tratarmos da linha principal estrada, também foram discutidos os ramais Diamantin...

We consider only finite undirected graphs without loops or multiple edges. For terminology and notation not defined here we refer the reader to Bondy and Murty’s textbook [6]. In this note we investigate irregularity measures of graphs. A graph is regular if all its vertices have the same degree, otherwise it is irregular. However, it is of interest to measure how irregular it is. Several appro...

In a graph G of maximum degree ∆ let γ denote the largest fraction of edges that can be ∆ edge-coloured. Albertson and Haas showed that γ ≥ 13 15 when G is cubic . We show here that this result can be extended to graphs with maximum degree 3 with the exception of a graph on 5 vertices. Moreover, there are exactly two graphs with maximum degree 3 (one being obviously the Petersen graph) for whic...

A spanning tree T of a graph G is called a homeomorphically irreducible spanning tree (HIST) if T does not contain vertices of degree 2. A graph G is called locally connected if for every vertex v ∈ V (G), the subgraph induced by the neighborhood of v is connected. In this paper, we prove that every connected and locally connected graph with more than 3 vertices contains a HIST. Consequently, w...

Albertson conjectured that if graph G has chromatic number r, then the crossing number of G is at least that of the complete graph Kr. This conjecture in the case r = 5 is equivalent to the four color theorem. It was verified for r = 6 by Oporowski and Zhao. In this paper, we prove the conjecture for 7 ≤ r ≤ 12 using results of Dirac; Gallai; and Kostochka and Stiebitz that give lower bounds on...