Let $G$ be any connected graph on $n$ vertices, $n \ge 2.$ Let $k$ be any positive integer. Suppose that a fire breaks out at some vertex of $G.$ Then, in each turn firefighters can protect at most $k$ vertices of $G$ not yet on fire; Next the fire spreads to all unprotected neighbours. The $k$-surviving rate of G, denoted by $ρ_k(G),$ is the expected fraction of vertices that can be saved from the fire, provided that the starting vertex is chosen uniformly at random.
In this note, it is shown that for any planar graph $G$ with average degree $4\frac{1}{2} - ε,$ where $ε\in (0, 1],$ we have $ρ_2(G) \ge \frac{2}{9}ε$. In particular, the result implies a significant improvement of the bound for 2-surviving rate for triangle-free planar graphs (Esperet, van den Heuvel, Maffray and Sipma, 2013) and for planar graphs without 4-cycles (Kong, Wang, Zhang, 2012). The proof is done using the separator theorem for planar graphs.
This paper is the corrected version of (Gordinowicz, 2018) unified with the corrigendum.
Let $G$ be any connected graph on $n$ vertices, $n \ge 2.$ Let $k$ be any positive integer. Suppose that a fire breaks out on some vertex of $G.$ Then in each turn $k$ firefighters can protect vertices of $G$ --- each can protect one vertex not yet on fire; Next a fire spreads to all unprotected neighbours. The \emph{$k$-surviving} rate of G, denoted by $ρ_k(G),$ is the expected fraction of vertices that can be saved from the fire by $k$ firefighters, provided that the starting vertex is chosen uniformly at random. In this paper, it is shown that for any planar graph $G$ we have $ρ_3(G) \ge \f…
In a graph $G$, a fire starts at some vertex. At every time step, firefighters can protect up to $k$ vertices, and then the fire spreads to all unprotected neighbours. The $k$-surviving rate $ρ_k(G)$ of $G$ is the expectation of the proportion of vertices that can be saved from the fire, if the starting vertex of the fire is chosen uniformly at random. For a given class of graphs $\cG$ we are interested in the minimum value $k$ such that $ρ_k(G)\geε$ for some constant $ε>0$ and all $G\in\cG$ i.e., such that linearly many vertices are expected to be saved in every graph from $\cG$). In this not…
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 Wa…
Studying the shortness of longest cycles in maximal planar graphs, we improve the upper bound on the shortness exponent of the class of $\frac{5}{4}$-tough maximal planar graphs presented by Harant and Owens [Discrete Math. 147 (1995), 301--305]. In addition, we present two generalizations of a similar result of Tkáč who considered $1$-tough maximal planar graphs [Discrete Math. 154 (1996), 321--328]; we remark that one of these generalizations gives a tight upper bound. We fix a problematic argument used in the first paper.
A graph is called 1-planar if it can be drawn in the plane so that each of its edges is crossed by at most one other edge. In this paper we decompose the set of all 1-planar graphs into three classes $\mathcal C_0, \mathcal C_1$ and $\mathcal C_2$ with respect to the types of crossings and present the decomposition of 1-planar join products. Zhang \cite{z} proved that every $n$-vertex 1-planar graph of class $\mathcal C_1$ has at most $\frac{18}{5}n-\frac{36}{5}$ edges and a $\mathcal C_1$-drawing with at most $\frac 35 n-\frac 65$ crossings. We improve these results. We show that every $\math…
A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. A graph, together with a 1-planar drawing is called 1-plane. Brandenburg et al. showed that there are maximal 1-planar graphs with only $\frac{45}{17}n + O(1)\approx 2.647n$ edges and maximal 1-plane graphs with only $\frac{7}{3}n+O(1)\approx 2.33n$ edges. On the other hand, they showed that a maximal 1-planar graph has at least $\frac{28}{13}n-O(1)\approx 2.15n-O(1)$ edges, and a maximal 1-plane graph has at least $2.1n-O(1)$ edges. We improve both lower bounds to $\frac{20n}{9}\approx 2.22n$.
We study the following Steinberg-type problem on circular coloring: for an odd integer $k\ge 3$, what is the smallest number $f(k)$ such that every planar graph of girth $k$ without cycles of length from $k+1$ to $f(k)$ admits a homomorphism to the odd cycle $C_k$ (or equivalently, is circular $(k,\frac{k-1}{2})$-colorable). Known results and counterexamples on Steinberg's Conjecture indicate that $f(3)\in\{6,7\}$. In this paper, we show that $f(k)$ exists if and only if $k$ is an odd prime. Moreover, we prove that for any prime $p\ge 5$, $$p^2-\frac{5}{2}p+\frac{3}{2}\le f(p)\le 2p^2+2p-5.$$ …
For positive integers $a$ and $b$, a graph $G$ is $(a:b)$-choosable if, for each assignment of lists of $a$ colors to the vertices of $G,$ each vertex can be colored with a set of $b$ colors from its list so that adjacent vertices are colored with disjoint sets. We show that for positive integers $a$ and $b$, every bipartite planar graph is $(a:b)$-choosable iff $\frac{a}{b} \ge 3$. For general planar graphs, we show that if $\frac{a}{b} < 4\frac{2}{5}$, then there exists a planar graph that is not $(a:b)$-choosable, thus improving on a result of X. Zhu, which had $4\frac{2}{9}$. Lastly, we sh…
A graph G is (a:b)-colorable if there exists an assignment of b-element subsets of {1,...,a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We first show that for every triangle-free planar graph G and a vertex x of G, the graph G has a set coloring phi by subsets of {1,...,6} such that |phi(v)|>=2 for each vertex v of G and |phi(x)|=3. As a corollary, every triangle-free planar graph on n vertices is (6n:2n+1)-colorable. We further use this result to prove that for every Delta, there exists a constant M_Delta such that every planar graph G of girth at least five a…
It is well-known that there exists a triangle-free planar graph of $n$ verticess such that the largest induced forest has size at most $\frac{5n}{8}$. Salavatipour proved that there is a forest of size at least $\frac{5n}{9.41}$ in any triangle-free planar graph of $n$ vertices. Dross, Montassier and Pinlou improved Salavatipour's bound to $\frac{5n}{9.17}$. In this work, we further improve the bound to $\frac{5n}{9}$. Our technique is inspired by the recent ideas from Lukot'ka, Maz{á}k and Zhu.
The concept of metric dimension has applications in a variety of fields, such as chemistry, robotic navigation, and combinatorial optimization. We show bounds for graphs with $n$ vertices and metric dimension $β$. For Hamiltonian outerplanar graphs, we have $β\leq \left\lceil\frac{n}2\right\rceil$; for outerplanar graphs in general, we have $β\leq \left\lfloor\frac{2n}{3}\right\rfloor$; for maximal planar graphs, we have $β\leq \left\lfloor\frac{3n}{4}\right\rfloor$. We also show that bipyramids have a metric dimension of $\left\lfloor\frac{2n}{5}\right\rfloor + 1$. It is conjectured that the …
Aichholzer et al. [{\it Graphs and Combinatorics}, Vol. 23, 481-507, 2007] introduced the notion of pseudo-convex partitioning of planar point sets and proved that the pseudo-convex partition number $ψ(n)$ satisfies, $\frac{3}{4}\lfloor\frac{n}{4}\rfloor\leq ψ(n)\leq\lceil\frac{n}{4}\rceil$. In this paper we prove that $ψ(13)=3$, which immediately improves the upper bound on $ψ(n)$ to $\lceil\frac{3n}{13}\rceil$, thus answering a question posed by Aichholzer et al. in the same paper.
In 1968, Gallai conjectured that the edges of any connected graph with $n$ vertices can be partitioned into $\lceil \frac{n}{2} \rceil$ paths. We show that this conjecture is true for every planar graph. More precisely, we show that every connected planar graph except $K_3$ and $K_5^-$ ($K_5$ minus one edge) can be decomposed into $\lfloor \frac{n}{2} \rfloor$ paths.
The crossing number $cr(G)$ of a graph $G=(V,E)$ is the smallest number of edge crossings over all drawings of $G$ in the plane. For any $k\ge 1$, the $k$-planar crossing number of $G$, $cr_k(G)$, is defined as the minimum of $cr(G_0)+cr(G_1)+\ldots+cr(G_{k-1})$ over all graphs $G_0, G_1,\ldots, G_{k-1}$ with $\cup_{i=0}^{k-1}G_i=G$. It is shown that for every $k\ge 1$, we have $cr_k(G)\le \left(\frac{2}{k^2}-\frac1{k^3}\right)cr(G)$. This bound does not remain true if we replace the constant $\frac{2}{k^2}-\frac1{k^3}$ by any number smaller than $\frac1{k^2}$. Some of the results extend to th…
We show that every planar graph $G$ has a 2-fold 9-coloring. In particular, this implies that $G$ has fractional chromatic number at most $\frac92$. This is the first proof (independent of the 4 Color Theorem) that there exists a constant $k<5$ such that every planar $G$ has fractional chromatic number at most $k$.
The 4 Color Theorem (4CT) implies that every $n$-vertex planar graph has an independent set of size at least $\frac{n}4$; this is best possible, as shown by the disjoint union of many copies of $K_4$. In 1968, Erdős asked whether this bound on independence number could be proved more easily than the full 4CT. In 1976 Albertson showed (independently of the 4CT) that every $n$-vertex planar graph has an independent set of size at least $\frac{2n}9$. Until now, this remained the best bound independent of the 4CT. Our main result improves this bound to $\frac{3n}{13}$.
In 1977, Wegner conjectured that the chromatic number of the square of every planar graph $G$ with maximum degree $Δ\ge8$ is at most $\bigl\lfloor\frac32Δ\bigr\rfloor+1$. We show that it is at most $\frac32 Δ(1+o(1))$ (where the $o(1)$ is as $Δ\to+\infty$), and indeed that this is true for the list chromatic number and for more general classes of graphs.
A graph $G$ is \emph{uniquely k-colorable} if the chromatic number of $G$ is $k$ and $G$ has only one $k$-coloring up to permutation of the colors. A uniquely $k$-colorable graph $G$ is edge-critical if $G-e$ is not a uniquely $k$-colorable graph for any edge $e\in E(G)$. Mel'nikov and Steinberg [L. S. Mel'nikov, R. Steinberg, One counterexample for two conjectures on three coloring, Discrete Math. 20 (1977) 203-206] asked to find an exact upper bound for the number of edges in a edge-critical 3-colorable planar graph with $n$ vertices. In this paper, we give some properties of edge-critical u…
We study the topic of "extremal" planar graphs, defining $\mathrm{ex_{_{\mathcal{P}}}}(n,H)$ to be the maximum number of edges possible in a planar graph on $n$ vertices that does not contain a given graph $H$ as a subgraph. In particular,we examine the case when $H$ is a small cycle,obtaining $\mathrm{ex_{_{\mathcal{P}}}}(n,C_{4}) \leq \frac{15}{7}(n-2)$ for all $n \geq 4$ and $\mathrm{ex_{_{\mathcal{P}}}}(n,C_{5}) \leq \frac{12n-33}{5}$ for all $n \geq 11$, and showing that both of these bounds are tight.
This paper proves that for each positive integer $m$, there is a planar graph $G$ which is not $(4m+\lfloor \frac{2m-1}{9}\rfloor,m)$-choosable. Then we pose some conjectures concerning multiple list colouring of planar graphs.
This paper proves that for each positive integer $m$, there is a triangle-free planar graph $G$ which is not $(3m+ \lceil \frac m{17} \rceil-1, m)$-choosable.
We prove that the Wiener Index $W(G)$ of a Maximal Planar graph $G$ with $n$ vertices satisfies $W(G) \leq \Big{\lfloor} \frac{1}{18}(n^3 + 3n^2) \Big{\rfloor}$ for $3 \leq n \leq 18$.
A matching of a graph is a set of edges without common end vertex. A graph is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. Recently, Biedl and Wittnebel proved that every 1-planar graph with minimum degree 3 and $n\geq 7$ vertices has a matching of size at least $\frac{n+12}{7}$, which is tight for some graphs. They also provided tight lower bounds for the sizes of matchings in 1-planar graphs with minimum degree 4 or 5. In this paper, we show that any 1-planar graph with minimum degree 6 and $n \geq 36$ vertices has a matching of size at lea…
The \emph{slow-coloring game} is played by Lister and Painter on a graph $G$. Initially, all vertices of $G$ are uncolored. In each round, Lister marks a nonempty set $M$ of uncolored vertices, and Painter colors a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. The score of the game is the sum of the sizes of all sets marked by Lister. The goal of Painter is to minimize the score, while Lister tries to maximize it. We provide strategies for Painter on various classes of graphs whose vertices can be partitioned into a bounded number of sets inducing fores…
A feedback vertex set of a graph is a subset of vertices intersecting all cycles. We provide tight upper bounds on the size of a minimum feedback vertex set in planar graphs of girth at least five. We prove that if $G$ is a connected planar graph of girth at least five on $n$ vertices and $m$ edges, then $G$ has a feedback vertex set of size at most $\frac{2m-n+2}{7}$. By Euler's formula, this implies that $G$ has a feedback vertex set of size at most $\frac{m}{5}$ and $\frac{n-2}{3}$. These results not only improve a result of Dross, Montassier and Pinlou and confirm the girth-5 case of one o…
We prove that every triangle-free planar graph of order $n$ and size $m$ has an induced linear forest with at least $\frac{9n - 2m}{11}$ vertices, and thus at least $\frac{5n + 8}{11}$ vertices. Furthermore, we show that there are triangle-free planar graphs on $n$ vertices whose largest induced linear forest has order $\lceil \frac{n}{2} \rceil + 1$.
We prove that the total curvature of a planar graph with nonnegative combinatorial curvature is at least $\frac{1}{12}$ if it is positive. Moreover, we classify the metric structures of ambient polygonal surfaces for planar graphs attaining this bound.
Let $G$ be a finite graph with minimum degree $r$. Form a random subgraph $G_p$ of $G$ by taking each edge of $G$ into $G_p$ independently and with probability $p$. We prove that for any constant $ε>0$, if $p=\frac{1+ε}{r}$, then $G_p$ is non-planar with probability approaching 1 as $r$ grows. This generalizes classical results on planarity of binomial random graphs.
Let ${\rm ex}_{\mathcal{P}}(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex planar graph which does not contain $H$ as a subgraph. When $T=K_2$, ${\rm ex}_{\mathcal{P}}(n,T,H)$ is the well studied function, the planar Turán number of $H$, denoted by ${\rm ex}_{\mathcal{P}}(n,H)$. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both ${\rm ex}_{\mathcal{P}}(n,C_4)$ and ${\rm ex}_{\mathcal{P}}(n,C_5)$. Later on, Y. Lan, et al. continued this topic and proved that ${\rm ex}_{\mathcal{P}}(n,C_6)\leq \frac{18(n-2)}{7}$. In thi…
The Wiener index of a connected graph is the sum of the distances between all pairs of vertices in the graph. It was conjectured that the Wiener index of an $n$-vertex maximal planar graph is at most $\lfloor\frac{1}{18}(n^3+3n^2)\rfloor$. We prove this conjecture and for every $n$, $n \geq 10$, determine the unique $n$-vertex maximal planar graph for which this maximum is attained.
Let $G$ be a $\{C_4, C_5\}$-free planar graph with a list assignment $L$. Suppose a preferred color is given for some of the vertices. We prove that if all lists have size at least four, then there exists an $L$-coloring respecting at least a constant fraction of the preferences.
Square coloring is a variant of graph coloring where vertices within distance two must receive different colors. When considering planar graphs, the most famous conjecture (Wegner, 1977) states that $\frac32Δ+1$ colors are sufficient to square color every planar graph of maximum degree $Δ$. This conjecture has been proven asymptotically for graphs with large maximum degree. We consider here planar graphs with small maximum degree and show that $2Δ+7$ colors are sufficient, which improves the best known bounds when $6\leqslant Δ\leqslant 31$.
In this paper, we study the flexibility of two planar graph classes $\mathcal{H}_1$, $\mathcal{H}_2$, where $\mathcal{H}_1$, $\mathcal{H}_2$ denote the set of all hopper-free planar graphs and house-free planar graphs, respectively. Let $G$ be a planar graph with a list assignment $L$. Suppose a preferred color is given for some of the vertices. We prove that if $G\in \mathcal{H}_1$ or $G\in \mathcal{H}_2$ such that all lists have size at least $5$, then there exists an $L$-coloring respecting at least a constant fraction of the preferences.
A planar graph is essentially $4$-connected if it is 3-connected and every of its 3-separators is the neighborhood of a single vertex. Jackson and Wormald proved that every essentially 4-connected planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{2n+4}{5}$, and this result has recently been improved multiple times. In this paper, we prove that every essentially 4-connected planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{5}{8}(n+2)$. This improves the previously best-known lower bound $\frac{3}{5}(n+2)$.
In 1979, Nishizeki and Baybars showed that every planar graph with minimum degree 3 has a matching of size $\frac{n}{3}+c$ (where the constant $c$ depends on the connectivity), and even better bounds hold for planar graphs with minimum degree 4 and 5. In this paper, we investigate similar matching-bounds for {\em 1-planar} graphs, i.e., graphs that can be drawn such that every edge has at most one crossing. We show that every 1-planar graph with minimum degree 3 has a matching of size at least $\frac{1}{7}n+\frac{12}{7}$, and this is tight for some graphs. We provide similar bounds for 1-plana…
This paper studies the strong fractional choice number $ch^s_f(G)$ and the strong fractional paint number $χ^s_{f,P}(G)$ of a graph $G$. We prove that these parameters of any finite graph are rational numbers. On the other hand, for any positive integers $p,q$ satisfying $2 \le \frac{2p}{2q+1} \leq \lfloor\frac{p}{q}\rfloor$, there exists a graph $G$ with $ch^s_f(G) = χ^s_{f,P}(G) = \frac{p}{q}$. The relationship between $χ^s_{f,P}(G)$ and $ch^s_f(G)$ is explored. We prove that the gap $χ^s_{f,P}(G)-ch^s_f(G)$ can be arbitrarily large. The strong fractional choice number of a family $\mathcal{…
We give here new upper bounds on the size of a smallest feedback vertex set in planar graphs with high girth. In particular, we prove that a planar graph with girth $g$ and size $m$ has a feedback vertex set of size at most $\frac{4m}{3g}$, improving the trivial bound of $\frac{2m}{g}$. We also prove that every $2$-connected graph with maximum degree $3$ and order $n$ has a feedback vertex set of size at most $\frac{n+2}{3}$.
A planar 3-connected graph $G$ is called \emph{essentially $4$-connected} if, for every 3-separator $S$, at least one of the two components of $G-S$ is an isolated vertex. Jackson and Wormald proved that the length $\mathop{\rm circ}\nolimits(G)$ of a longest cycle of any essentially 4-connected planar graph $G$ on $n$ vertices is at least $\frac{2n+4}{5}$ and Fabrici, Harant and Jendrol' improved this result to $\mathop{\rm circ}\nolimits(G)\geq \frac{1}{2}(n+4)$. In the present paper, we prove that an essentially 4-connected planar graph on $n$ vertices contains a cycle of length at least $\…
We give here some new lower bounds on the order of a largest induced forest in planar graphs with girth $4$ and $5$. In particular we prove that a triangle-free planar graph of order $n$ admits an induced forest of order at least $\frac{6n+7}{11}$ , improving the lower bound of Salavatipour [M. R. Salavatipour, Large induced forests in triangle-free planar graphs, Graphs and Combinatorics, 22:113-126, 2006]. We also prove that a planar graph of order $n$ and girth at least $5$ admits an induced forest of order at least $\frac{44n+50}{69}$.
A graph is $k$-degenerate if every subgraph has minimum degree at most $k$. We provide lower bounds on the size of a maximum induced 2-degenerate subgraph in a triangle-free planar graph. We denote the size of a maximum induced 2-degenerate subgraph of a graph $G$ by $α_2(G)$. We prove that if $G$ is a connected triangle-free planar graph with $n$ vertices and $m$ edges, then $α_2(G) \geq \frac{6n - m - 1}{5}$. By Euler's Formula, this implies $α_2(G) \geq \frac{4}{5}n$. We also prove that if $G$ is a triangle-free planar graph on $n$ vertices with at most $n_3$ vertices of degree at most thre…
It is well-known that every $n$-vertex planar graph with minimum degree 3 has a matching of size at least $\frac{n}{3}$. But all proofs of this use the Tutte-Berge-formula for the size of a maximum matching. Hence these proofs are not directly algorithmic, and to find such a matching one must apply a general-purposes maximum matching algorithm, which has run-time $O(n^{1.5}α(n))$ for planar graphs. In contrast to this, this paper gives a linear-time algorithm that finds a matching of size at least $\frac{n}{3}$ in any planar graph with minimum degree 3.
An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an \emph{interval $t$-coloring} if all colors are used and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. In 1990, Kamalian proved that if a graph $G$ with at least one edge has an interval $t$-coloring, then $t\leq 2|V(G)|-3$. In 2002, Axenovich improved this upper bound for planar graphs: if a planar graph $G$ admits an interval $t$-coloring, then $t\leq \frac{11}{6}|V(G)|$. In the same paper Axenovich suggested a conjecture that if a planar graph $G$ has an interval $t$-colo…
We introduce the fractional version of oriented coloring and initiate its study. We prove some basic results and study the parameter for directed cycles and sparse planar graphs. In particular, we show that for every $ε> 0$, there exists an integer $g_ε \geq 12$ such that any oriented planar graph having girth at least $g_ε$ has fractional oriented chromatic number at most $4+ε$. Whereas, it is known that there exists an oriented planar graph having girth at least $g_ε$ with oriented chromatic number equal to $5$. We also study the fractional oriented chromatic number of directed cycles and pr…
