A step toward Chen-Lih-Wu conjecture

TL;DR

The paper advances equitable $k$-coloring under Ore-type degree-sum constraints by proving that for every fixed $c>0$, all sufficiently large $n$ and $k\ge cn$ satisfy the Chen–Lih–Wu conjecture and its Ore-type relaxation, with a polynomial-time algorithm to decide equitable $k$-colorability in this regime. The approach blends absorption methods, Szemerédi Regularity with blow-up techniques, and $K_r$-tiling results, reducing the problem to finding tilings in reduced graphs and balanced multipartite graphs. A key contribution is the segmentation into non-extremal and extremal cases, handling each via absorbers and partition-based tilings, and proving that extremal configurations either occur or yield a $K_r$-factor. The results significantly strengthen the link between equitable coloring conjectures and extremal graph structure, providing both theoretical and algorithmic consequences with potential extensions to degree-sequence variants. This work thus delivers a near-resolution of the conjectures in the large-$n$, linear-in-$k$ regime under Ore-type bounds, with implications for related coloring and tiling problems in sparse and structured graphs.

Abstract

An equitable $k$-coloring of a graph is a proper $k$-coloring where the sizes of any two different color classes differ by at most one. In 1973, Meyer conjectured that every connected graph $G$ has an equitable $k$-coloring for some $k\leq Δ(G)$, unless $G$ is a complete graph or an odd cycle. Chen, Lih, and Wu strengthened this in 1994 by conjecturing that for $k\geq 3$, the only connected graphs of maximum degree at most $k$ with no equitable $k$-coloring are the complete bipartite graph $K_{k,k}$ for odd $k$ and the complete graph $K_{k+1}$. A more refined conjecture was proposed by Kierstead and Kostochka, relaxing the maximum degree condition to an Ore-type condition. Their conjecture states the following: for $k\geq 3$, if $G$ is an $n$-vertex graph such that $d(x) + d(y)\leq 2k$ for every edge $xy\in E(G)$, and $G$ admits no equitable $k$-coloring, then $G$ contains either $K_{k+1}$ or $K_{m,2k-m}$ for some odd $m$. We prove that for any constant $c>0$ and all sufficiently large $n$, the latter two conjectures hold for every $k\geq cn$. Our proof yields an algorithm with polynomial time that decides whether $G$ has an equitable $k$-coloring, thereby answering a conjecture of Kierstead, Kostochka, Mydlarz, and Szemerédi when $k \ge cn$.

Figures (7)

• Figure 1: (EX2). Each thick line indicates that all possible edges exist between the two corresponding parts.
• Figure 2: A $K_3$-absorber for $\{u_1,u_2,u_3\}$.
• Figure 3: $|N_R(K_t)\cap V(K_r)|\geq r-t+1$ for $r=3$ and $t=2$.
• Figure 4: $|N_R(K_t)\cap V(K_r)|\leq r-t$ for $r=3$ and $t=2$.
• Figure 5: Vertices $x_j,y_j$ exchanged in the $k$-th step, where $\delta=\beta+(k-1)\alpha$ and $D \in \mathcal{P}_{k-1}\setminus{A_k^{k-1}}$.

Theorems & Definitions (72)

• Conjecture 1.1: Chen-Lih-Wu Conjecture CLW1994
• Theorem 1.2: KK2008
• Conjecture 1.3: KK2008
• Conjecture 1.4: KKMS2008
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Lemma 2.2: Graph Removal Lemma ADLR1994
• Proposition 2.3
• proof