The potential-function conjecture for critical graphs

Fix an integer $k\geq 6$. For a graph $G$, let

$$T(G)=\max\{2a(H)+b(H):H\subseteq G\text{ is a union of vertex-disjoint cliques}\},$$

where $a(H)$ is the number of components of $H$ isomorphic to $K_{k-1}$ and $b(H)$ is the number of components of $H$ isomorphic to $K_{k-2}$. For positive real numbers $\varepsilon$ and $\delta$, define the $(\varepsilon,\delta)$-potential by

$$p(G)=((k-2)(k+1)+\varepsilon)|V(G)|-2(k-1)|E(G)|-\delta T(G).$$

Potential-function conjecture. For every $k\geq 6$, there exist $\varepsilon_k,\delta_k,P_k>0$ such that the $(\varepsilon_k,\delta_k)$-potential satisfies

$$p(K_k)=k(k-3)+k\varepsilon_k-2\delta_k,$$

$\displaystyle p(G)\leq k(k-3)+|V(G)|\varepsilon_k-\left(2+\frac{|V(G)|-1}{k-1}\right)\delta_k$ if $G$ is $k$-Ore and $G\neq K_k$, and

$$p(G)\leq k(k-3)-P_k$$

if $G$ is $k$-critical and not $k$-Ore. Here a graph is $k$-critical if $\chi(G)=k$ and every proper subgraph has chromatic number less than $k$, and $k$-Ore graphs are the graphs obtained from $K_k$ by repeated Ore-compositions. This conjecture proposes a potential gap for non-$k$-Ore critical graphs while prescribing the potential of $K_k$ and bounding it on nontrivial $k$-Ore graphs; it is intended to provide the structural and discharging framework for the unresolved cases $6\leq k\leq 32$.