The Total Choosability with Neighbor Sum Distinguishing Properties of Planar Graphs That Are Devoid of \(C_5\) Adjacent to \(C_3\)

Abstract

Let $G$ be a graph such that a proper total coloring $\psi:V(G)\cup E(G)\longrightarrow \mathbb{N}$.  Let $s(x)$ denote the sum of colors assigned to $x$ and those incident edges of $x$.  A coloring $\psi$ is a \emph{neighbor sum distinguishing total coloring} if $s(x)\neq s(y)$, whenever $xy$ is an edge in $G$. Let $L$ be a $k$-list assignment of a graph $G$ if $L$ is a function, say $L:V(G)\cup F(G)\rightarrow 2^{\mathbb{N}}$ such that $L(t)=k$ for all  $t$ in the set $V(G)\cup F(G.)$ If $G$ has a total coloring such that $\alpha(x)$ is the element of $ L(x)$ for all $x$ in the set $ V(G)\cup E(G)$, then we call $\alpha$ {\it total-$L$-coloring}. Moreover, a total-$L$-coloring $\alpha$ is called a  \emph{neighbor sum distinguishing total-}$L$\emph{-coloring} if $s(x)\neq s(y)$ where $xy$ is an edge in $G$.  Let $Ch''_{\sum}(G)$ be the minimum number $k$ such that $G$ has such coloring for every $k$-list assignment $L$. \indent In this work, we present that for a graph $G$ has $Ch''_{\sum}(G)$  $\leq\max\{10,\Delta(G)+3\}$ if $G$ is a planar graph that are devoid of  $C_5$ adjacent to $C_3$.