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For a planar graph $H$, let $operatorname{mathbf{N}}_{mathcal P}(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. In this paper, we prove that $operatorname{mathbf{N}}_{mathcal P}(n,P_7)sim{4over 27}n^4$, $operatorname{mathbf{N}}_{mathcal P}(n,C_6)sim(n/3)^3$, $operatorname{mathbf{N}}_{mathcal P}(n,C_8)sim(n/4)^4$ and $operatorname{mathbf{N}}_{mathcal P}(n,K_4{1})sim(n/6)^6$, where $K_4{1}$ is the $1$-subdivision of $K_4$. In addition, we obtain significantly improved upper bounds on $operatorname{mathbf{N}}_{mathcal P}(n,P_{2m+1})$ and $operatorname{mathbf{N}}_{mathcal P}(n,C_{2m})$ for $mgeq 4$. For a wide class of graphs $H$, the key technique developed in this paper allows us to bound $operatorname{mathbf{N}}_{mathcal P}(n,H)$ in terms of an optimization problem over weighted graphs.
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Hakimi, Schmeichel, and Thomassen in 1979 conjectured that every $4$-connected planar triangulation $G$ on $n$ vertices has at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. In this paper, we show that every $4$-connected planar triangulation on $n$ vertices has $Omega(n^2)$ Hamiltonian cycles. Moreover, we show that if $G$ is a $4$-connected planar triangulation on $n$ vertices and the distance between any two vertices of degree $4$ in $G$ is at least $3$, then $G$ has $2^{Omega(n^{1/4})}$ Hamiltonian cycles.
The problem of maximising the number of cliques among $n$-vertex graphs from various graph classes has received considerable attention. We investigate this problem for the class of $1$-planar graphs where we determine precisely the maximum total number of cliques as well as the maximum number of cliques of any fixed size. We also precisely characterise the extremal graphs for these problems.
In 1930, Kuratowski showed that $K_{3,3}$ and $K_5$ are the only two minor-minimal non-planar graphs. Robertson and Seymour extended finiteness of the set of forbidden minors for any surface. v{S}ir{a}v{n} and Kochol showed that there are infinitely many $k$-crossing-critical graphs for any $kge 2$, even if restricted to simple $3$-connected graphs. Recently, $2$-crossing-critical graphs have been completely characterized by Bokal, Oporowski, Richter, and Salazar. We present a simplified description of large 2-crossing-critical graphs and use this simplification to count Hamiltonian cycles in such graphs. We generalize this approach to an algorithm counting Hamiltonian cycles in all 2-tiled graphs, thus extending the results of Bodrov{z}a-Pantic, Kwong, Doroslovav{c}ki, and Pantic for $n = 2$.
Let $mathscr{G}$ be the class of plane graphs without triangles normally adjacent to $8^{-}$-cycles, without $4$-cycles normally adjacent to $6^{-}$-cycles, and without normally adjacent $5$-cycles. In this paper, it is showed that every graph in $mathscr{G}$ is $3$-choosable. Instead of proving this result, we directly prove a stronger result in the form of weakly DP-$3$-coloring. The main theorem improves the results in [J. Combin. Theory Ser. B 129 (2018) 38--54; European J. Combin. 82 (2019) 102995]. Consequently, every planar graph without $4$-, $6$-, $8$-cycles is $3$-choosable, and every planar graph without $4$-, $5$-, $7$-, $8$-cycles is $3$-choosable. In the third section, it is proved that the vertex set of every graph in $mathscr{G}$ can be partitioned into an independent set and a set that induces a forest, which strengthens the result in [Discrete Appl. Math. 284 (2020) 626--630]. In the final section, tightness is considered.
In 1976, Steinberg conjectured that planar graphs without $4$-cycles and $5$-cycles are $3$-colorable. This conjecture attracted numerous researchers for about 40 years, until it was recently disproved by Cohen-Addad et al. (2017). However, coloring planar graphs with restrictions on cycle lengths is still an active area of research, and the interest in this particular graph class remains. Let $G$ be a planar graph without $4$-cycles and $5$-cycles. For integers $d_1$ and $d_2$ satisfying $d_1+d_2geq8$ and $d_2geq d_1geq 2$, it is known that $V(G)$ can be partitioned into two sets $V_1$ and $V_2$, where each $V_i$ induces a graph with maximum degree at most $d_i$. Since Steinbergs Conjecture is false, a partition of $V(G)$ into two sets, where one induces an empty graph and the other induces a forest is not guaranteed. Our main theorem is at the intersection of the two aforementioned research directions. We prove that $V(G)$ can be partitioned into two sets $V_1$ and $V_2$, where $V_1$ induces a forest with maximum degree at most $3$ and $V_2$ induces a forest with maximum degree at most $4$; this is both a relaxation of Steinbergs conjecture and a strengthening of results by Sittitrai and Nakprasit (2019) in a much stronger form.
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