For an integer $k\geq2$, let $\chi_k'(G)$ denote the minimum number of colors in an edge-coloring of a graph $G$ such that every nonzero degree in each color subgraph is congruent to $1\pmod{k}$. A graph is a $0_k$-graph if every vertex degree is divisible by $k$. We disprove a conjecture of Berthe et al.\ (On modular edge colorings of graphs, SIAM J. Discrete Math. 40 (2026) 897--904), which states that $\chi_k'(G)\leq k+o(k)$ for every $0_k$-graph $G$. We prove a lower bound for $0_k$-graphs with degree set $\{k,2k\}$ and a specified vertex partition. With a suitable choice of the part sizes, if the number of edges inside one part is $o(k^2)$, then $\chi_k'(G)\geq(4-2\sqrt2+o(1))k$. This gives connected bipartite and connected nonbipartite counterexamples. In particular, the same examples also disprove the earlier conjecture of Botler, Colucci, and Kohayakawa (The mod $k$ chromatic index of graphs is $O(k)$, J. Graph Theory 102 (2023) 197--200), which states that $\chi_k'(G)\leq k+C$ for some absolute constant $C$.
Let $k\geq2$ be an integer. A $1\bmod k$ edge-coloring of a graph $G$ is an edge-coloring in which every nonzero degree in each color class is congruent to $1$ modulo $k$. Let $\chi'_k(G)$ denote the minimum number of colors required, and let $\chi'_k$ be the supremum of $\chi'_k(G)$ over all finite simple graphs $G$. Botler, Colucci, and Kohayakawa conjectured that there exists an absolute constant $C$ such that $\chi'_k(G)\leq k+C$ for every $k$ and every $G$. We disprove this conjecture, even within the class of bipartite graphs. More precisely, for all integers $c\geq0$ and $k\geq3c+2$, we construct a finite simple bipartite graph $G_{k,c}$ satisfying $\chi'_k(G_{k,c})=k+c+1$. Consequently, $\chi'_k\geq k+\lfloor(k+1)/3\rfloor$ for every $k\geq2$. For $k_m=2\cdot3^{m-1}$, we give an affine-hyperplane construction of a finite simple bipartite graph $G_m$ satisfying $\Delta(G_m)=\chi'_{k_m}(G_m)=3^m=3k_m/2$. More generally, for every sufficiently large $k$, we construct a finite simple bipartite graph $G_k$ such that $\Delta(G_k)=\chi'_k(G_k)\geq3k/2-10(k\log k)^{1/3}$. Our proofs combine a codegree obstruction with explicit cyclic and affine-geometric constructions and a structured random perturbation.
A $(k,d)$-edge coloring of a loopless multigraph $G$ is an edge coloring using at most $k$ colors such that the subgraph formed by each color class has maximum degree at most $d$. The least such $k$ is denoted by $\chi'_d(G)$. Let $G$ be a loopless non-bipartite multigraph with maximum degree $\Delta(G)$ and odd girth $g_0(G)$, and let $d\ge1$ be odd. We prove that \[ \chi'_d(G)\le\left\lceil\frac{g_0(G)\Delta(G)-1}{dg_0(G)-1}\right\rceil. \] For $d=1$, this is Goldberg's odd-girth refinement of Shannon's theorem, while for $g_0(G)=3$ it is the defective Shannon bound of Aboulker, Aubian, and Huang. For every odd $d>1$, every odd $g_0\ge3$, and every $\Delta>d$, an almost full ring multigraph $R(\Delta,g_0)$, an odd cycle with edge multiplicities alternating between $\lfloor\Delta/2\rfloor$ and $\lceil\Delta/2\rceil$, except that two consecutive edges have multiplicity $\lfloor\Delta/2\rfloor$, attains equality. We also derive a range in which the defective Goldberg--Seymour conjecture holds.
Let $G$ be a simple graph with maximum degree $\Delta\ge 3$, and let $P(G,k)$ denote its chromatic polynomial. For each positive integer $k$, the list-color function $P_{\ell}(G,k)$ is the minimum number of $L$-colorings of $G$ over all $k$-assignments $L$. In this paper, we prove that $P_{\ell}(G,k)=P(G,k)$ for every integer $k\ge 23.41\Delta$. This gives a threshold for equality that is linear in the maximum degree and independent of the number of vertices or edges. It improves the known sufficient condition $k\ge |E(G)|-1$ for graphs with sufficiently many edges relative to their maximum degree.
A famous conjecture of Erd\H{o}s and Hajnal (1969) states that for every integer $g\ge 4$ there is a smallest function $f_g:\mathbb{N}\to\mathbb{N}$ such that every graph of chromatic number at least $f_g(k)$ contains a subgraph of chromatic number $k$ and girth at least $g$. So far, this has only been proved for $g=4$ by R\"odl (1977), with $f_4(k)$ bounded by a tower of height $\Theta(k^2\log k)$. We exhibit a surprising connection between finding high-chromatic subgraphs of large odd-girth (avoiding short odd cycles) and lower-bounding multicolor Ramsey numbers of odd cycles. Using this connection, we prove that for every odd $g\ge 5$ there is a function $h_g:\mathbb{N}\to\mathbb{N}$ growing as a power tower of height $\frac{g-3}{2}$ such that every graph of chromatic number at least $h_g(k)$ contains a subgraph of chromatic number at least $k$ and odd-girth at least $g$. This proves a conjecture of Mohar and Wu (2018), addresses a question of Erd\H{o}s and Hajnal (1975), and for $g=5$ improves R\"odl's bound on $f_4(k)$ to a single-exponential. We extend this to a much more general meta-theorem which applies to many graph parameters: if $f$ is the fractional chromatic number, the Hall ratio, or the strict vector chromatic number (Lov\'{a}sz-Theta-function of the complement), then for every $k,g\in\mathbb{N}$, every graph $G$ with sufficiently large $f(G)$ contains a subgraph $G'$ of odd-girth at least $g$ with $f(G')\ge k$. The key Ramsey-theoretic ingredient is a new lower bound on Ramsey numbers of odd cycles. For $p\ge 1$, let $\mathcal{O}_p=\{C_3,C_5,\ldots,C_{2p+1}\}$. We show that $R_k(\mathcal{O}_p)\ge(\log^{(p-1)}k)^{k/3-o(k)}$ for every fixed $p$, where $\log^{(p-1)}$ denotes the $(p-1)$-fold iterated logarithm. This yields the first superexponential lower bound on multicolor Ramsey numbers of fixed odd cycles, and extends the recent breakthrough by OpenAI for triangles.
A B-coloring of a graph $G$ is a proper edge-coloring in which every $4$-cycle receives four distinct colors; let $q_B(G)$ be the minimum number of colors in such a coloring. Every graph of maximum degree $\Delta$ is $K_{2,\Delta+1}$-free; hence the known $2\Delta$ bound for planar graphs with $\Delta\ge38$ (Kong et al., 2026) motivates our study of $K_{2,t}$-free planar graphs, where $t\ge2$ is an integer. We prove $q_B(G)=\Delta(G)$ when $t=2$ and $\Delta(G)\ge7$, or when $t\ge3$ and $\Delta(G)\ge14(t-1)$. For $t\ge35$, the bound $q_B(G)\le\Delta(G)+t-1$ holds regardless of $\Delta(G)$; for every $t\ge2$, it also holds when $\Delta(G)>428$. Finally, for every integer $k\ge1$, every $k$-degenerate $K_{2,t}$-free graph satisfies $q_B(G)\le\Delta(G)+(k-1)\min\{t-1,\Delta(G)\}$, with equality for $K_{k,t-1}$ when $k\ge2$ and $t-1\ge k$.
For a graph $G$, a proper edge coloring of $G$ is called a D-coloring if every diamond subgraph of $G$ is rainbow. Let $\chi'_D(G)$ be the D-chromatic index of $G$, which is the smallest integer $k$ such that $G$ admits a D-coloring with $k$ colors. Let $\Delta$ be the maximum degree of $G$. The only known Brooks-type upper bound on $\chi'_D(G)$ is $\frac{9}{16}\Delta^2 + \frac{1}{2}\Delta$, given by a greedy coloring. In this paper, using a probabilistic method, we obtain the first improvement upon this upper bound by proving that $\chi'_D(G) \le (1-c)\frac{9}{16}\Delta^2$ for some $c>0$ and sufficiently large $\Delta$.
Lin Tian, Run-Ze Wang· 0 citations
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