On the Edge-Difference and Edge-Sum Chromatic Sum of the Simple Graphs

Document Type : Research Paper


School of Mathematical Science, Shahrood University of Technology, Shahrood, Iran.



‎For a coloring $c$ of a graph $G$‎, ‎the edge-difference coloring sum and edge-sum coloring sum with respect to the coloring $c$ are respectively‎ ‎$\sum_c D(G)=\sum |c(a)-c(b)|$ and $\sum_s S(G)=\sum (c(a)+c(b))$‎, ‎where the summations are taken over all edges $ab\in E(G)$‎.
‎The edge-difference chromatic sum‎, ‎denoted by $\sum D(G)$‎, ‎and the edge-sum chromatic sum‎, ‎denoted by $\sum S(G)$‎, ‎are respectively the minimum possible values‎ ‎of $\sum_c D(G)$ and $\sum_c S(G)$‎, ‎where the minimums are taken over all proper coloring of $c$‎.
‎In this work‎, ‎we study the edge-difference chromatic sum and the edge-sum chromatic sum of graphs‎. ‎In this regard‎,
‎we present some necessary conditions for the existence of homomorphism between two graphs‎. ‎Moreover‎, ‎some upper and lower bounds for these parameters in terms of the fractional chromatic number are introduced‎
‎as well‎.


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