Adjacent vertex distinguishing total coloring of the corona product of graphs

An adjacent vertex distinguishing (AVD-)total coloring of a simple graph G is a proper total coloring of G such that for any pair of adjacent vertices u and v, we haveC(u)6=C(v), where C(u) is the set of colors given to vertex u and the edges incident tou for u is an element of V(G). The AVD-total chromatic number,chi '' a(G), of a graph G is the minimum number of colors required foran AVD-total coloring ofG. The AVD-total coloring conjecture states thatfor any graphGwith maximum degree triangle, chi '' a(G)<=triangle+3. The total coloringconjecture states that for any graphGwith maximum degree triangle,chi ''(G)<=triangle+2, where chi ''(G) is the total chromatic number ofG, that is, the minimumnumber of colors needed for a proper total coloring ofG. A graphGissaid to be AVD-total colorable(total colorable), ifGsatisfies the AVD-totalcoloring conjecture (total coloring conjecture). In this paper, we prove thatfor any AVD-total colorable graphGand any total-colorable graphHwith triangle(H)<=triangle(G), the corona productG degrees HofGandHsatisfies the AVD-totalcoloring conjecture. We also prove that the graphG degrees Knadmits an AVD-total coloring using (triangle(G degrees Kn) +p) colors, if there is an AVD-total coloringof graphGusing (triangle(G) +p) colors, where p is an element of{1,2,3}. Furthermore, givena total colorable graphGand positive integer r and p where 1 <= p <= 3, weclassify the corona graphsG(r)=G degrees G degreesdegrees G(r+ 1 times) such that chi '' a(G(r)) = triangle(G((r))) +p.