Chromatic support bridge

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Proved bounds, not a contradiction. Use the least-budget minimal-ratio core, with

$$|H|=ar+1,\quad \alpha(H)=a,\quad \delta(H)\ge r, \quad S:=\sum_v(d_H(v)-r)\le rh,\quad h=r-a,$$

where $r\ge12$ and $a\le r-5$. Every nonempty independent set obeys

$$|N_H[I]|\ge r|I|+1. \tag{1}$$

Choose an inclusion-minimal vertex set $U$ such that $J=H[U]$ is not $r$-colorable, and put $m=|U|$, $b=\alpha(J)$. Such a set exists because $|H|>r\alpha(H)$. Then

$$\boxed{b\le\min\{a,h+1\},\qquad (r-2)m\le r^2(h+1)-3r-2.} \tag{2}$$

If $J=H$, the stronger conclusion is

$$\boxed{a\le\lfloor(r+1)/2\rfloor.} \tag{3}$$

The distinction between vertex-critical $J$ and an independence-critical core is essential throughout.

A bridge using the supplied density theorem

Let $c=e_H(U,V(H)\setminus U)$. Nonnegativity of degree surplus gives

$$S\ge2e(J)+c-rm. \tag{4}$$

For a maximum independent $b$-set $I$ of $J$, inducedness makes $I$ independent in $H$. By (1),

$$c\ge |N_H[I]\setminus U|\ge rb+1-m. \tag{5}$$

An edge-minimal non-$r$-colorable subgraph of $J$ must be spanning: otherwise its vertex support contradicts the choice of $U$. It is therefore $(r+1)$-critical in the usual subgraph sense. The supplied Kostochka--Yancey source, Theorem 3 and equation (9) gives

$$2e(J)-rm\ge(1-2/r)m-r+1+2/r. \tag{6}$$

The digest was read before checking that passage. Combining (4)--(6) with $S\le rh$ proves

$$rb\le r(h+1)-2-2/r+2m/r, \tag{7}$$

and hence $b\le h+2$. This already supplies independence control on the chromatic support, rather than merely a scalar order estimate. Here $b$ is the independence number of the induced graph $J$, not of its edge-pruned critical subgraph.

The non-Ore strengthening

The previously proved Ore-cap exclusion says that every $(r+1)$-Ore graph satisfying the $(r+1)$-set cap has at least $r^2+1$ vertices. It applies to the spanning critical subgraph just extracted, since edge deletion preserves the upper caps. That subgraph has at most $ar+1<r^2+1$ vertices, so is non-Ore.

An additional primary source gives a useful strict improvement: Kostochka and Yancey, A Brooks-type result for sparse critical graphs, Theorem 6, printed page 4. For a non-Ore $k$-critical graph, $k\ge6$, it states

$$2e\ge\frac{(k+1)(k-2)m-(k^2-5k+2)}{k-1}.$$

Its statement was checked in the primary text; its full proof was not independently rederived. Substituting $k=r+1$ strengthens (6) to

$$2e(J)-rm\ge(1-2/r)m-r+3+2/r. \tag{8}$$

Using $c\ge0$ in (4) gives the order bound in (2). Using (5) instead gives

$$\boxed{rb\le r(h+1)-4-2/r+2m/r.} \tag{9}$$

To obtain $b\le h+1$, first suppose $h+1<r/2$. The order bound in (2) implies

$$m<r^2/2+2r+1,$$

because substituting this latter expression into its left side exceeds $r^3/2-3r-2$ by $r^2$. Equation (9) and integrality then give the claim. If $h+1\ge r/2$, the claim already follows from $b\le a$ unless $r$ is even and $a=r/2+1=h+2$. In that sole case $m\le ar+1=r^2/2+r+1<r^2/2+2r+1$, so (9) again suffices. This proves (2).

If $J=H$, substitute $b=a$, $m=ar+1$ into (9). It gives

$$2a(r-1)\le r^2+r-4, \qquad a\le(r+2)/2-1/(r-1),$$

and hence (3).

Actual color coupling strengthens the support cap

The core still carries an actual coloring in which every foreign color has independence at most $r$. If a foreign color graph on $U$ could partition into $r$ cliques, those same parts would be independent in $J$, contradicting $\chi(J)>r$. Thus the elementary nonpartite Turan gap applies to each foreign color. For $m\ge2r+1$, it gives

$$\boxed{e(J)\le D_r(m)-(r-1)(\lfloor m/r\rfloor-1).} \tag{10}$$

Indeed every foreign color has at least $p_r(m)+\lfloor m/r\rfloor-1$ edges, and their edge sets partition the complement of $J$ on $U$. This is an upper bound for this chromatic support, not an inherited bound on every induced subgraph.

What remains missing

Even with (2), $J$ need not have $m>rb$. The hereditary span theorem therefore cannot be applied merely because $b\le h+1$ is smaller than $a$. No theorem has been established saying that every capped graph of small independence number is $r$-colorable below its capacity $rb$. Likewise (3) applies only if the entire core is vertex-critical; the core has not been proved to have that property.

The expansion/cut bridge, the substitution in the supplied density theorem, and the non-Ore strengthening against the later primary theorem were checked by the author; the argument is author-recorded. These estimates do not establish an unrestricted infinite family of positive cases.