A vertex-coloring of a graph is centered if every connected subgraph has a vertex with a unique color. A vertex-coloring of a graph is linear if every path in the graph has a vertex with a unique color. Let $\chi_{\mathrm{cen}}(G)$ and $\chi_{\mathrm{lin}}(G)$ be the minimum number of colors in a centered (resp. linear) coloring of $G$. We present a family of graphs witnessing that if $f$ is a nondecreasing function such that $\chi_{\mathrm{cen}}(G) \leq f(\chi_{\mathrm{lin}}(G))$ for every graph $G$, then $f(k) = \Omega(k^2 / \log k)$. The construction was found by OpenAI's GPT-5.6 Sol Pro.
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...
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...
For a graph $G$, let $h(G)$ be the minimum cardinality of a vertex set meeting every maximum independent set of $G$. We establish two complementary reduction principles for the Bollob\'as--Erd\H{o}s--Tuza conjecture: the conjecture for arbitrary graphs is equivalent to its restriction to regular graphs of any fixed pos...
We study graph coloring with color preferences, in which each vertex ranks the available colors. In addition to assigning different colors to adjacent vertices, we require the coloring to be stable: no group of vertices can cyclically exchange their assigned colors so that each strictly prefers its new color to its ori...
Tomohiro Koana, Y. Oh, Hirotaka Yoneda· 0 citations
A majority C-coloring of a graph $G$ assigns colors to the vertices such that every vertex shares its color with at least half of its neighbors. The maximum number of colors that can be used in such a coloring of $G$ is denoted by $\overline{\chi}_{\geqslant}(G)$. In this paper, the focus is on the majority C-coloring...
Csilla Bujtás, M. Dettlaff, Hanna Furmanczyk et al.· 0 citations
A path in a properly edge-colored graph is rainbow if its edges have pairwise distinct colors. For a proper edge-coloring $c$ of a graph $G$, let $\operatorname{rpc}(G,c)$ be the minimum number of rainbow paths needed to cover $E(G)$, and let $\operatorname{rpc}(G)$ be the maximum of $\operatorname{rpc}(G,c)$ over all...
Xiao-Chuan Liu, Bo-Yan Xu, Xu Yang· 0 citations
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