We prove that every $n$-vertex planar graph $G$ with no triangle sharing an edge with a 4-cycle has independence ratio $n/α(G) \leq 4 - \varepsilon$ for $\varepsilon = 1/30$. This result implies that the same bound holds for 4-cycle-free planar graphs and planar graphs with no adjacent triangles and no triangle sharing an edge with a 5-cycle. For the latter case we strengthen the bound to $\varepsilon = 2/9$.
The independence equivalence class of a graph $G$ is the set of graphs that have the same independence polynomial as $G$. Beaton, Brown and Cameron (arXiv:1810.05317) found the independence equivalence classes of even cycles, and raised the problem of finding the independence equivalence class of odd cycles. The problem is completely solved in this paper.
Results on independent sets in categorical products of graphs, the ultimate categorical independence ratio and the ultimate categorical independent domination ratio
DP-coloring was introduced by Dvořák and Postle as a generalization of list coloring. It was originally used to solve a longstanding conjecture by Borodin, stating that every planar graph without cycles of lengths 4 to 8 is 3-choosable. In this paper, we give three sufficient conditions for a planar graph to be DP-4-colorable. Actually all the results (Theorem 1.3, 1.4 and 1.7) are stated in the ``color extendability'' form, and uniformly proved by vertex identification and discharging method.
A polyhedral graph is a $3$-connected planar graph. We find the least possible order $p(k,a)$ of a polyhedral graph containing a $k$-independent set of size $a$ for all positive integers $k$ and $a$. In the case $k = 1$ and $a$ even, we prove that the extremal graphs are exactly the vertex-face (radial) graphs of maximal planar graphs.
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 shown 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…
We prove Engström's conjecture that the independence complex of graphs with no induced cycle of length divisible by $3$ is either contractible or homotopy equivalent to a sphere. Our result strengthens a result by Zhang and Wu, verifying a conjecture of Kalai and Meshulam which states that the total Betti number of the independence complex of such a graph is at most $1$. A weaker conjecture was proved earlier by Chudnovsky, Scott, Seymour, and Spirkl, who showed that in such a graph, the number of independent sets of even size minus the number of independent sets of odd size has values $0$, $1…
We show that the choosability of the square of planar graphs of max degree 4 without five cycles is at most 12. Keywords: planar graph, choosability AMS Mathematics Subject Classification: 05C15
Whitney proved in 1931 that every 4-connected planar triangulation is hamiltonian. Later in 1979, Hakimi, Schmeichel and Thomassen conjectured that every such triangulation on $n$ vertices has at least $2(n - 2)(n - 4)$ hamiltonian cycles. Along this direction, Brinkmann, Souffriau and Van Cleemput established a linear lower bound on the number of hamiltonian cycles in 4-connected planar triangulations. In stark contrast, Alahmadi, Aldred and Thomassen showed that every 5-connected triangulation of the plane or the projective plane has exponentially many hamiltonian cycles. This gives the moti…
The concept of DP-coloring of a graph is a generalization of list coloring introduced by Dvořák and Postle in 2015. Multiple DP-coloring of graphs, as a generalization of multiple list coloring, was first studied by Bernshteyn, Kostochka and Zhu in 2019. This paper proves that planar graphs without 3-cycles and normally adjacent 4-cycles are $(7m, 2m)$-DP-colorable for every integer $m$. As a consequence, the strong fractional choice number of any planar graph without 3-cycles and normally adjacent 4-cycles is at most $7/2$.
The induced odd cycle packing number $iocp(G)$ of a graph $G$ is the maximum integer $k$ such that $G$ contains an induced subgraph consisting of $k$ pairwise vertex-disjoint odd cycles. Motivated by applications to geometric graphs, Bonamy et al.~\cite{indoc} proved that graphs of bounded induced odd cycle packing number, bounded VC dimension, and linear independence number admit a randomized EPTAS for the independence number. We show that the assumption of bounded VC dimension is not necessary, exhibiting a randomized algorithm that for any integers $k\ge 0$ and $t\ge 1$ and any $n$-vertex g…
A graph is near-bipartite if its vertex set can be partitioned into an independent set and a set which induces a forest. In this paper, planar graphs without cycles of length from 4 to 7 are shown to be near-bipartite.
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 $Ω(n^2)$ 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 $2^{Ω(n^{1/4})}$ Hamiltonian cycles.
In this paper, we show that every planar graph without $4$-cycles and $6$-cycles has a partition of its vertex set into two sets, where one set induces a forest, and the other induces a forest with maximum degree at most $2$ (equivalently, a disjoint union of paths). Note that we can partition the vertex set of a forest into two independent sets. However, a pair of independent sets combined may not induce a forest. Thus our result extends the result of Wang and Xu (2013) stating that the vertex set of every planar graph without $4$-cycles and $6$-cycles can be partitioned into three sets, wher…
DP-3-coloring of planar graphs without $4,9$-cycles and two cycles from $\{5,6,7,8\}$
列表着色的一个推广，现在称为DP着色，最近由Dvopelák和Postle引入。本质上，DP着色在邻顶点的颜色列表之间分配任意匹配，而不是像列表着色那样仅匹配相同的颜色。关于可平面图列表染色的一些结果已经被推广到DP染色的情形。我们注意到列表着色结果并不总是扩展到DP着色结果。本文的主要结果是证明了每个不含长为$\{4, a, b, 9\}$ for $a, b \in \{6, 7, 8\}$的圈的平面图都是DP-$3$-可着色的，推广了关于可平面图的$3$ -可选性的三个已有结果.
Hu and Li investigate the signed graph version of Erd$\ddot{\mathrm{o}}$s problem: Is there a constant $c$ such that every signed planar graph without $k$-cycles, where $4\leq k\leq c$, is $3$-colorable and prove that each signed planar graph without cycles of length from 4 to 8 is 3-colorable. We give a very short and simple proof of this result and improve it, based on a recent observation.
We resolve a conjecture of Cox and Martin by determining asymptotically for every $k\ge 2$ the maximum number of copies of $C_{2k}$ in an $n$-vertex planar graph.
Whitney proved in 1931 that 4-connected planar triangulations are Hamiltonian. Hakimi, Schmeichel, and Thomassen conjectured in 1979 that if $G$ is a 4-connected planar triangulation with $n$ vertices then $G$ contains at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. On the other hand, a recent result of Alahmadi, Aldred, and Thomassen states that there are exponentially many Hamiltonian cycles in 5-connected planar triangulations. In this paper, we consider 4-connected planar $n$-vertex triangulations $G$ that do not have too many separating 4-cyc…
In the 1960s, Erdős and his cooperators initiated the research of the maximum numbers of edges in a graph or a planar graph on $n$ vertices without $k$ edge-disjoint cycles. This problem had been solved for $k\leq4$. As pointed out by Bollobás, it is very difficult for general $k$. Recently, Tait and Tobin [J. Combin. Theory Ser. B, 2017] confirmed a famous conjecture on maximum spectral radius of $n$-vertex planar graphs. Motivated by the above results, we consider two extremal problems on planar graphs without $k$ edge-disjoint cycles. We first determine the maximum number of edges in a plan…
