Let G be a graph. A minimal coloring of G is a coloring which
has the smallest possible sum among all proper colorings of G,
using natural numbers. The vertex-strength of G, denoted by
s(G), is the minimum number of colors which is necessary to
obtain a minimal coloring. In this note we study these concepts,
and define a new concept called the edge-strength of G, denoted
by s′(G). We pose some upper bounds for s(G) using ∆(G) and col(G). Also, it is proved that s′(G) lies between
∆(G) and ∆(G)+1, but it may not be equal to
X′(G). Based on our results on vertex-strength, we conjecture
that:
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s(G) ≤ ⎡\dfracX (G)+∆(G)2⎤. |
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