Independence number and packing coloring of generalized Mycielski graphs

Abstract

A simple graph is called semisymmetric if it is regular and edge-transitive but not vertex-transitive. Let p be an arbitrary prime. Folkman proved [Regular line-symmetric graphs, J. Combin. Theory 3 (1967) 215-232] that there is no semisymmetric graph of order 2p or 2p 2 . In this paper an extension of his result in the case of cubic graphs of order 20p 2 is given. We prove that there is no connected cubic semisymmetric graph of order 20p 2 or, equivalently, that every connected cubic edge-transitive graph of order 20p 2 is necessarily symmetric.

Gamma coloring of Mycielskian graphs

“…. Also, it has been proved in (5) that for a graph G, γ (µ (G)) = γ (G) + 1 and therefore γ (µ(P n )) =…”

“…Let C be a proper coloring of a graph G. A subset U of V(G) is said to be C-colorful if every vertex of U receives different color under the coloring C. A subset U of V(G) is said to be color transversal with respect to the coloring C if U intersects every color class of the coloring C. The study of a graph theoretical parameter in Mycielskian construction of graphs is one of the interesting research fields in graph theory. Some of such studies are Dominator coloring of Mycielskian graphs (1) , strong coloring of Mycielskian graphs (2) , connectivity of Mycielskian graphs (3,4) , packing coloring of Mycielskian graphs (5) , total chromatic number of Mycielskian graphs (6) , diameter of Mycielskian of graphs (7) , total weight choosability of Mycielskian graphs (8) and Hamilton-connected Mycielskian graphs (9,10) . In this paper we define the notion of gamma coloring and discuss about gamma coloring of Mycielskian graphs.…”

A characterization of 4-$χ_S$-vertex-critical graphs for $S\succeq (1,3,3,3,\ldots)$

“…Proof. Since H 14 and H 15 are 4-χ S ′ -vertex-critical, where S ′ ∈ S 1,3,4 , and (1, 3, 4) (1,3,3), it suffices to show that χ S (H 14 ) = χ S (H 15 ) = 4. The patten "4 1 2 3 1 1 1" is an S-packing 4-coloring of H 14 , so χ S (H 14 ) ≤ 4.…”

“…The development on the packing chromatic number up to 2020 has been summarized in the substantial survey [6]. Research into this concept is still flourishing, the developments after the survey include [1,2,5,8,10].…”

“…If S 1 = (s 1 1 , s and G admits an S 2 -packing k-coloring, then, clearly, G also admits an S 1 -packing k-coloring. In [11,Theorem 3.1], Gastineau proved the following appealing dichotomy result: If S is a packing sequence with |S| = 4, then the decision problem whether a given graph G admits an S-packing coloring is polynomial-time solvable if S S ′ , where S ′ ∈ { (2,3,3,3), (2,2,3,4), (1,4,4,4), (1,2,5,6)}, and NP-complete otherwise.…”