VERTEX-STRENGTH OF FUZZY GRAPHS

NEW

Problem: Reconstructed statement: for a finite Rosenfeld fuzzy graph $G=(V,\sigma,\mu)$, with

and with $s(G)$ the Eslahchi–Onagh chromatic strength, Conjecture 2.14 asserts:

and, for every integer $k\ge2$, some fuzzy graph $G_k$ satisfies $k-1\le\Delta(G_k)$ and $s(G_k)=\Delta(G_k)+k$.

Result: The literal upper bound is false.

Let $V=\{a,b,c\}$, set $\sigma(v)=1/4$ for all $v$, and set

Then $G$ is a valid fuzzy graph, since $\mu(u,v)=\min(\sigma(u),\sigma(v))$. Every pair of vertices is a full/strong edge, so a fuzzy coloring must assign the three vertices to three distinct color classes. Hence every chromatic-sum-optimal coloring uses exactly three colors, and

But each vertex has degree

so $\Delta(G)=1/2$. Thus

while $s(G)=3$, contradicting $s(G)\le2\Delta(G)+1$.

The second clause is actually consistent: for $k\ge2$, take the complete fuzzy graph on $2k-1$ vertices with all vertex and edge memberships $1/2$. Then $\Delta=k-1$ and $s=2k-1=\Delta+k$.

Diagnosis: the defect is the missing lower bound on memberships of color-conflict edges. If every such edge has membership at least $1/2$, then the usual recoloring argument gives the repaired bound: in an optimal coloring, a vertex colored $r$ must have a neighbor in each lower color $1,\dots,r-1$; each contributes at least $1/2$ to its fuzzy degree, so $r-1\le2d(v)\le2\Delta(G)$.

Citation: No prior counterexample is used. The reconstruction follows the definitions in Eslahchi–Onagh, “Vertex-strength of fuzzy graphs,” IJMMS 2006, and the standard Rosenfeld fuzzy graph degree convention.