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Automorphism groups and Distinguishing Colorings of Central and Middle Graphs
Amitayu Banerjee, Alexa Gopaulsingh, Zalán Molnár
Let G be a simple, finite, connected, and undirected graph. The middle graph M(G) of G is obtained from the subdivision graph S(G) after joining pairs of subdivided vertices that l…
On the Cost Number of Graphs with Determining Number Two
Alexa Gopaulsingh, Zalán Molnár, Amitayu Banerjee
A distinguishing vertex coloring of a graph is a vertex coloring such that only the identity automorphism of preserves the coloring. A graph is -distinguishable if it ad…
Distinguishing chromatic number of middle and subdivision graphs
Amitayu Banerjee, Alexa Gopaulsingh, Zalán Molnár
Let be a simple finite connected graph of order greater than or equal to . We obtain the following results: (1). We apply a result of Hamada and Yoshimura from 1976 and…
Brooks' type theorems for coloring parameters of locally finite graphs and Konig's Lemma
Amitayu Banerjee, Zalán Molnár, Alexa Gopaulsingh
In the past, analogues to Brooks' theorem have been found for various parameters of graph coloring for infinite locally finite connected graphs in ZFC. We prove these theorems are…
Upper bounds for the list-distinguishing chromatic number
Amitayu Banerjee, Zalán Molnár, Alexa Gopaulsingh
We prove analogs of Brooks' Theorem for the list-distinguishing chromatic number of different classes of simple finite connected graphs. Moreover, we determine two upper bounds for…
Distinguishing colorings, proper colorings, and covering properties without the Axiom of Choice
Amitayu Banerjee, Zalán Molnár, Alexa Gopaulsingh
We work with simple graphs in ZF (Zermelo--Fraenkel set theory without the Axiom of Choice (AC)) and assume that the sets of colors can be either well-orderable or non-well-orderab…