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The vertex arboricity $a(G)$ of a graph $G$ is the minimum $k$ such that $V(G)$ can be partitioned into $k$ sets where each set induces a forest. For a planar graph $G$, it is known that $a(G)leq 3$. In two recent papers, it was proved that planar graphs without $k$-cycles for some $kin{3, 4, 5, 6, 7}$ have vertex arboricity at most 2. For a toroidal graph $G$, it is known that $a(G)leq 4$. Let us consider the following question: do toroidal graphs without $k$-cycles have vertex arboricity at most 2? It was known that the question is true for k=3, and recently, Zhang proved the question is true for $k=5$. Since a complete graph on 5 vertices is a toroidal graph without any $k$-cycles for $kgeq 6$ and has vertex arboricity at least three, the only unknown case was k=4. We solve this case in the affirmative; namely, we show that toroidal graphs without 4-cycles have vertex arboricity at most 2.
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We initiate a systematic study of the fractional vertex-arboricity of planar graphs and demonstrate connections to open problems concerning both fractional coloring and the size of the largest induced forest in planar graphs. In particular, the following three long-standing conjectures concern the size of a largest induced forest in a planar graph, and we conjecture that each of these can be generalized to the setting of fractional vertex-arboricity. In 1979, Albertson and Berman conjectured that every planar graph has an induced forest on at least half of its vertices, in 1987, Akiyama and Watanabe conjectured that every bipartite planar graph has an induced forest on at least five-eighths of its vertices, and in 2010, Kowalik, Luv{z}ar, and v{S}krekovski conjectured that every planar graph of girth at least five has an induced forest on at least seven-tenths of its vertices. We make progress toward the fractional generalization of the latter of these, by proving that every planar graph of girth at least five has fractional vertex-arboricity at most $2 - 1/324$.
Arboricity is a graph parameter akin to chromatic number, in that it seeks to partition the vertices into the smallest number of sparse subgraphs. Where for the chromatic number we are partitioning the vertices into independent sets, for the arboricity we want to partition the vertices into cycle-free subsets (i.e., forests). Arboricity is NP-hard in general, and our focus is on the arboricity of cographs. For arboricity two, we obtain the complete list of minimal cograph obstructions. These minimal obstructions do generalize to higher arboricities; however, we no longer have a complete list, and in fact, the number of minimal cograph obstructions grows exponentially with arboricity. We obtain bounds on their size and the height of their cotrees. More generally, we consider the following common generalization of colouring and partition into forests: given non-negative integers $p$ and $q$, we ask if a given cograph $G$ admits a vertex partition into $p$ forests and $q$ independent sets. We give a polynomial-time dynamic programming algorithm for this problem. In fact, the algorithm solves a more general problem which also includes several other problems such as finding a maximum $q$-colourable subgraph, maximum subgraph of arboricity-$p$, minimum vertex feedback set and minimum $q$ of a $q$-colourable vertex feedback set.
A $k$-linear coloring of a graph $G$ is an edge coloring of $G$ with $k$ colors so that each color class forms a linear forest -- a forest whose each connected component is a path. The linear arboricity $chi_l(G)$ of $G$ is the minimum integer $k$ such that there exists a $k$-linear coloring of $G$. Akiyama, Exoo and Harary conjectured in 1980 that for every graph $G$, $chi_l(G)leq left lceil frac{Delta(G)+1}{2}rightrceil$ where $Delta(G)$ is the maximum degree of $G$. First, we prove the conjecture for 3-degenerate graphs. This establishes the conjecture for graphs of treewidth at most 3 and provides an alternative proof for the conjecture in some classes of graphs like cubic graphs and triangle-free planar graphs for which the conjecture was already known to be true. Next, for every 2-degenerate graph $G$, we show that $chi_l(G)=leftlceilfrac{Delta(G)}{2}rightrceil$ if $Delta(G)geq 5$. We conjecture that this equality holds also when $Delta(G)in{3,4}$ and show that this is the case for some well-known subclasses of 2-degenerate graphs. All our proofs can be converted into linear time algorithms.
All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constraining the structure of the graph, for any $ell in {3,4,5,6,7}$, a planar graph is 4-choosable if it is $ell$-cycle-free. In terms of constraining the list assignment, one refinement of $k$-choosability is choosability with separation. A graph is $(k,s)$-choosable if the graph is colorable from lists of size $k$ where adjacent vertices have at most $s$ common colors in their lists. Every planar graph is $(4,1)$-choosable, but there exist planar graphs that are not $(4,3)$-choosable. It is an open question whether planar graphs are always $(4,2)$-choosable. A chorded $ell$-cycle is an $ell$-cycle with one additional edge. We demonstrate for each $ell in {5,6,7}$ that a planar graph is $(4,2)$-choosable if it does not contain chorded $ell$-cycles.
Given a graph $G$, the strong clique number of $G$, denoted $omega_S(G)$, is the maximum size of a set $S$ of edges such that every pair of edges in $S$ has distance at most $2$ in the line graph of $G$. As a relaxation of the renowned ErdH{o}s--Nev{s}etv{r}il conjecture regarding the strong chromatic index, Faudree et al. suggested investigating the strong clique number, and conjectured a quadratic upper bound in terms of the maximum degree. Recently, Cames van Batenburg, Kang, and Pirot conjectured a linear upper bound in terms of the maximum degree for graphs without even cycles. Namely, if $G$ is a $C_{2k}$-free graph, then $omega_S(G)leq (2k-1)Delta(G)-{2k-1choose 2}$, and if $G$ is a $C_{2k}$-free bipartite graph, then $omega_S(G)leq kDelta(G)-(k-1)$. We prove the second conjecture in a stronger form, by showing that forbidding all odd cycles is not necessary. To be precise, we show that a ${C_5, C_{2k}}$-free graph $G$ with $Delta(G)ge 1$ satisfies $omega_S(G)leq kDelta(G)-(k-1)$, when either $kgeq 4$ or $kin {2,3}$ and $G$ is also $C_3$-free. Regarding the first conjecture, we prove an upper bound that is off by the constant term. Namely, for $kgeq 3$, we prove that a $C_{2k}$-free graph $G$ with $Delta(G)ge 1$ satisfies $omega_S(G)leq (2k-1)Delta(G)+(2k-1)^2$. This improves some results of Cames van Batenburg, Kang, and Pirot.
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