A disproof of a gap-one conjecture for the equitable chromatic number of block graphs

Abstract: For a graph $G$, let $$L(G)=\max{ω(G),\lceil (|V(G)|+1)/(α<em>{\min}(G)+1)\rceil}$$, where $ω(G)$ is the clique number and $α</em>{\min}(G)$ is the minimum, over all vertices $v$, of the largest size of an independent set containing $v$. Dybizbański, Furmańczyk, and Mkrtchyan (Discrete Appl. Math. 354 (2024), 15--28) conjectured that every block graph $G$ satisfies $L(G)\leqχ<em>{=}(G)\leq L(G)+1$, where $χ</em>{=}(G)$ is the equitable chromatic number of $G$. We disprove this conjecture in a strong form. For every pair of integers $d\geq 2$ and $k\geq 4d-1$, we construct a connected block graph $G_{d,k}$ such that $L(G_{d,k})=k$ and $χ<em>{=}(G</em>{d,k})=k+d$. Thus the difference $χ_{=}(G)-L(G)$ is unbounded on connected block graphs.

• The paper provides a disproof of the Dybizbański–Furmańczyk–Mkrtchyan gap-one conjecture for the equitable chromatic number of block graphs through construction of a graph family.
• The construction involves fixed $2$ block graph $G_{d, k}$ and demonstrates the gap between the equitable chromatic number and a defined lower bound can be unbalanced.
• The disproof shows that every connected block graph has no upper bound for the ratio between equitable chromatic number and lower bound and that the bounded example was graph $32$ and needed $9$ colors.

Background and context

An equitable $q$-coloring of a graph $G$ is a proper vertex coloring with $q$ colors whose color classes differ in size by at most one; the equitable chromatic number $\chi_{eq}(G)$ is the least such $q$. The classical Hajnal–Szemerédi theorem guarantees an equitable $(\Delta+1)$-coloring for every graph of maximum degree $\Delta$, but for restricted graph classes one seeks bounds in terms of structural parameters. For a vertex $v$, let $\alpha(G,v)$ denote the maximum size of an independent set containing $v$, and define

$G$0

Dybizbański, Furmańczyk, and Mkrtchyan observed that every graph satisfies

$G$1

since in any equitable $G$2-coloring, the class containing a vertex $G$3 with $G$4 has size at most $G$5 while all other classes have size at most $G$6, forcing $G$7. Their examples showed that this lower bound can be off by one color, and by analogy with Vizing–Goldberg-type phenomena they conjectured that every block graph $G$8 satisfies $G$9. Supporting evidence included forests (via Chang's theorem), well-covered block graphs, connected block graphs with $q$0, block graphs where each cut vertex lies in exactly two blocks, and exhaustive computation up to order 19.

The paper under discussion disproves this conjecture in the strongest possible sense.

Main result

The central theorem states that for every pair of integers $q$1 and $q$2 there exists a connected block graph $q$3 with $q$4 and $q$5. Taking $q$6 arbitrarily large yields the corollary that $q$7 is unbounded on connected block graphs: no bound of the form $q$8 with an absolute constant $q$9 can hold. This is a decisive refutation not only of the gap-one conjecture but of any fixed additive gap.

The construction

Fix $\chi_{eq}(G)$0, $\chi_{eq}(G)$1, and let $\chi_{eq}(G)$2. The graph $\chi_{eq}(G)$3 consists of a central clique $\chi_{eq}(G)$4 with distinguished vertices $\chi_{eq}(G)$5; two $\chi_{eq}(G)$6-blocks attached at $\chi_{eq}(G)$7 and two at $\chi_{eq}(G)$8 (their non-central vertices being pairwise disjoint "private" vertices); and a single pendant edge $\chi_{eq}(G)$9. This is a connected block graph of order

$q$0

a value engineered so that any equitable coloring with fewer than $q$1 colors must have minimum class size at least four. The assumption $q$2 ensures the attached blocks are maximum cliques, so $q$3.

Three lemmas establish the required values:

Small $q$4: An independent set containing $q$5 avoids all other vertices of $q$6, both blocks at $q$7, all but at most one private vertex per block at $q$8, plus possibly $q$9, giving $(\Delta+1)$0; symmetrically $(\Delta+1)$1. Every other vertex admits an independent set of size at least five, so $(\Delta+1)$2.

Lower-bound value: Combining $(\Delta+1)$3, $(\Delta+1)$4, and $(\Delta+1)$5 (valid since $(\Delta+1)$6), we obtain $(\Delta+1)$7 exactly.

No fewer than $(\Delta+1)$8 colors: The key obstruction is isolated in a general lemma. If adjacent vertices $(\Delta+1)$9 satisfy $\Delta$0 for some vertex $\Delta$1, then in any equitable $\Delta$2-coloring at least one of the two distinct classes of $\Delta$3 and $\Delta$4 misses $\Delta$5 and hence has size at most $\Delta$6, so $\Delta$7. Deleting the pendant $\Delta$8 drops $\Delta$9 (the pendant vertex was essential to reaching size four). Applying the lemma with $v$0, $v$1 gives $v$2 for any equitable $v$3-coloring, yet $v$4 forces $v$5 — a contradiction.

Upper coloring at $v$6 colors: Color $v$7 injectively with colors $v$8 ($v$9, $\alpha(G,v)$0, $\alpha(G,v)$1) and $\alpha(G,v)$2 with color 1. The private vertices of the four $\alpha(G,v)$3-blocks are colored bijectively from carefully chosen $\alpha(G,v)$4-subsets of $\alpha(G,v)$5 omitting designated colors near $\alpha(G,v)$6, $\alpha(G,v)$7, and $\alpha(G,v)$8, ensuring properness within blocks and compatibility across the shared clique. Class-size accounting shows exactly four classes of size 3 (colors $\alpha(G,v)$9, $v$0, $v$1, $v$2) and all others of size 4, matching $v$3.

Strength and sharpness of the counterexamples

Beyond unboundedness, the family is structurally minimal in several senses. Each $v$4 has exactly six blocks, three cut vertices, diameter 3, independence number 6, and $v$5; its block-cut-tree shape is fixed independently of $v$6 and $v$7, and no cut vertex belongs to more than three blocks. Since the gap $v$8 remains unbounded under all these restrictions, the disproof isolates precisely which hypothesis matters: the conjecture holds when every cut vertex lies in exactly two blocks, and permitting a cut vertex in three blocks already suffices for arbitrarily large gaps. This sharply delineates the boundary of validity of the gap-one phenomenon on block graphs.

The smallest explicit counterexample occurs at $v$9, $G$00: a graph on 32 vertices with $G$01 and $G$02, violating the conjectured $G$03 bound by two colors — notable given that exhaustive search had verified the conjecture up to order 19.

Limitations and open questions

The disproof leaves open whether a multiplicative bound holds: does there exist a constant $G$04 such that $G$05 for every connected block graph? The family provides partial information on any such constant: $G$06 forces $G$07, and along the extremal choice $G$08 the ratio $G$09 tends to $G$10, so $G$11 asymptotically. Whether even this multiplicative form fails remains unresolved. The construction also relies on the specific interplay between the pendant vertex and the two high-degree cut vertices $G$12; it does not address whether analogous gaps persist under further restrictions, e.g., bounded number of blocks or bounded degree of the block-cut-tree beyond what is already imposed here.

Conclusion

This note refutes the Dybizbański–Furmańczyk–Mkrtchyan gap-one conjecture via an explicit six-block family $G$13 achieving $G$14 for arbitrary $G$15, proving the gap between the natural lower bound and the equitable chromatic number is unbounded on connected block graphs. A clean deletion argument — removing a pendant vertex shrinks the largest independent sets through two adjacent cut vertices simultaneously — supplies the lower-bound mechanism, while a hand-crafted coloring achieves the exact upper value. The result narrows the valid scope of the gap-one phenomenon to subclasses excluding a cut vertex in three or more blocks, and frames the multiplicative analogue as the natural remaining question.