We study the parameterized complexity of $k$-Coloring in $H$-free graphs, when $H$ is a linear forest (i.e., a disjoint union of paths) as an induced subgraph. We show two hardness results: * $k$-Coloring is W[1]-hard in $2P_2$-free graphs when parameterized by $k$. * $3$-Coloring is W[1]-hard in $P_t$-free graphs when parameterized by $t$. Moreover, assuming the ETH, these problems admit no algorithms solving $n$-vertex instances in time $f(k) \cdot n^{o(k)}$ and $f(t) \cdot n^{o(t/\log t)}$, respectively, for any computable function $f$. The first result resolves in a strong form a long-standing open problem, originally posed by Ho\`ang, Kami\'nski, Lozin, Sawada, and Shu [Algorithmica, 2010]. The second result answers a question of Golovach, Johnson, Paulusma, and Song [Journal of Graph Theory, 2017].
This work generalizes and combines tools from the $(k+2)-coloring to $k$-list-coloring reduction of [Zamir, ICALP 2021] and the hypergraph-containers based approach in [Zamir, STOC 2023] and yields an iterable reduction from $(k+1)$-list-coloring to $k$-list-coloring over fixed palettes.
It is proved that PCF-COLORABILITY is NP-complete for bipartite graphs, and linear-time algorithms for PCF-COLORABILITY are provided in block graphs, proper interval graphs, chain graphs, and pseudo-split graphs.
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...
The main result shows that the exact average-case complexity of this fundamental problem is $\Theta(nk)$ for every $k \leq n^{c'}$ and some $c'\in (0, 1)$, and reveals the average sublinear nature of $k$-colorability: the average-case complexity is linear in $n$, and thus sublinear in the size of the input.
Cassandra Marcussen, Edward Pyne, R. Rubinfeld et al.· arXiv.org· 0 citations
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...
The first $poly(\Delta,\log n)-round algorithm for $(\Delta + 1)$-edge coloring in the CONGEST model is presented and the $n$-dependency of its runtime, $\tilde{O}(\log^5 n)$, matches the best published dependency in the LOCAL model.
Sebastian Brandt, Ananth Narayanan, Alexandre Nolin· 0 citations
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