Remove the maximum-degree restriction from the total 2-coalition bound

Prove that every graph G with minimum degree \(\delta(G)\ge 2\) and maximum degree \(\Delta(G)\) satisfies \(\mathrm{TC}_2(G)\le \left\lfloor\frac{\delta(G)}{2}\right\rfloor\bigl(\Delta(G)-2\left\lfloor\frac{\delta(G)}{2}\right\rfloor+1\bigr)+\left\lceil\frac{\delta(G)}{2}\right\rceil\), without assuming the restriction \(\Delta(G)\ge 4\left\lfloor\frac{\delta(G)}{2}\right\rfloor-2\).

Background

Theorem 3.4 establishes the stated upper bound for the total 2-coalition number TC2(G)\mathrm{TC}_2(G), but only for graphs whose maximum degree satisfies Δ(G)≥4⌊δ(G)/2⌋−2\Delta(G)\ge 4\lfloor\delta(G)/2\rfloor-2. The construction G(d,ℓ)G(d,\ell) shows that the bound is sharp for broad families, including graphs with arbitrary even minimum degree and sufficiently large maximum degree.

The authors report that they could not find a counterexample when the maximum-degree restriction is removed. They therefore formulate the unrestricted inequality as a conjecture. Establishing it would extend the principal upper-bound theorem to all graphs with minimum degree at least two.

References

It is possible that Theorem~\ref{thm:upperbound_delta} is true even if the restriction $\Delta(G)\ge 4\lfloor \delta(G)/2\rfloor-2$ is omitted. In spite of extensive investigations, we could not find a counterexample to that statement. In addition, we base our suspicion that the restriction can be omitted because this is true in the case when $\delta(G)\le 5$, which follows from Theorem~\ref{Combination}. Based on the above discussion, we propose the following:

If $G$ is a graph with $\delta=\delta(G)\ge 2$ and $\Delta=\Delta(G)$, then $$\TC_{2}(G)\leq \left\lfloor\frac{\delta}{2}\right\rfloor(\Delta-2\left\lfloor \frac{\delta}{2}\right\rfloor+1)+\left\lceil\frac{\delta}{2}\right\rceil.$$

— Total $k$-coalition: bounds, exact values and an application to double coalition  (2502.07310 - Brešar et al., 11 Feb 2025) in Section 3, immediately before the Conjecture following Example 1