Debarun Ghosh, Ervin Győri, Oliver Janzer, Addisu Paulos, Nika Salia, Oscar Zamora
Finding the maximum number of induced cycles of length $k$ in a graph on $n$ vertices has been one of the most intriguing open problems of Extremal Graph Theory. Recently Balogh, Hu, Lidický and Pfender answered the question in the case $k=5$. In this paper we determine precisely, for all sufficiently large $n$, the maximum number of induced $5$-cycles that an $n$-vertex planar graph can contain.
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We determine the maximum number of edges that a planar graph can have as a function of its maximum degree and matching number.
In 1979, Hakimi and Schmeichel considered the problem of maximizing the number of cycles of a given length in an $n$-vertex planar graph. They precisely determined the maximum number of triangles and $4$-cycles and presented a conjecture for the maximum number of pentagons. In this work, we confirm their conjecture. Even more, we characterize the $n$-vertex, planar graphs with the maximum number of pentagons.
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.
On the maximum spectral radius of planar graphs
本文研究了具有固定顶点数的平面图的最大谱半径，给出了基于阶数的紧致界，并证明了在包含支配顶点且阶数 $n \geq 48$ 的平面图中，$P_{2}$ 与 $P_{n-2}$ 的联图取得最大谱半径。
A graph is path-pairable if for any pairing of its vertices there exist edge-disjoint paths joining the vertices in each pair. We investigate the behaviour of the maximum degree in path-pairable planar graphs. We show that any $n$-vertex path-pairable planar graph must contain a vertex of degree linear in $n$.
Deciding whether a planar graph (even of maximum degree $4$) is $3$-colorable is NP-complete. Determining subclasses of planar graphs being $3$-colorable has a long history, but since Grötzsch's result that triangle-free planar graphs are such, most of the effort was focused to solving Havel's and Steinberg's conjectures. In this paper, we prove that every planar graph of maximum degree $4$ obtained as a subgraph of the medial graph of any bipartite plane graph is $3$-choosable. These graphs are allowed to have close triangles (even incident), and have no short cycles forbidden, hence represen…
2-Distance Coloring of Planar Graphs with Specific Maximum Degree
研究最大度数至少为6的平面图的2距离着色问题，探讨其着色方法及结果。
Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $\left\{1, \ldots, n\right\}$ with $m=m(n)$ edges. We show that in the sparse regime, when $\limsup_{n \to \infty} m/n<1$, with high probability the maximum degree of $P(n,m)$ takes at most two different values.
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.
Kühn, Osthus and Taraz showed that for each γ>0 there exists C such that any n-vertex graph with minimum degree γn contains a planar subgraph with at least 2n-C edges. We find the optimum value of C for all γ<1/2 and sufficiently large n.
We study squares of planar graphs with the aim to determine their list chromatic number. We present new upper bounds for the square of a planar graph with maximum degree $Δ\leq 4$. In particular $G^2$ is 5-, 6-, 7-, 8-, 12-, 14-choosable if the girth of $G$ is at least 16, 11, 9, 7, 5, 3 respectively. In fact we prove more general results, in terms of maximum average degree, that imply the results above.
We prove that the maximum degree $Δ_n$ of a random series-parallel graph with $n$ vertices satisfies $Δ_n/\log n \to c$ in probability, and $\mathbb{E}\, Δ_n \sim c \log n$ for a computable constant $c>0$. The same result holds for outerplanar graphs.
For large $n$ we determine exactly the maximum numbers of induced $C_4$ and $C_5$ subgraphs that a planar graph on $n$ vertices can contain. We show that $K_{2,n-2}$ uniquely achieves this maximum in the $C_4$ case, and we identify the graphs which achieve the maximum in the $C_5$ case. This extends work in a paper by Hakimi and Schmeichel and a paper by Ghosh, Győri, Janzer, Paulos, Salia, and Zamora which together determine both maxima asymptotically.
An \emph{equitable coloring} of a graph is a proper vertex coloring such that the sizes of every two color classes differ by at most 1. Chen, Lih, and Wu conjectured that every connected graph $G$ with maximum degree $Δ\geq 2$ has an equitable coloring with $Δ$ colors, except when $G$ is a complete graph or an odd cycle or $Δ$ is odd and $G=K_{Δ,Δ}.$ Nakprasit proved the conjecture holds for planar graphs with maximum degree at least 9. Zhu and Bu proved that the conjecture holds for every $C_3$-free planar graph with maximum degree at least 8 and for every planar graph without $C_4$ and $C_5$…
On Wegner's 8-Coloring Theorem for Planar Graphs of Maximum Degree Three
该研究简化了Wegner猜想特殊情况（最大度≤3的平面图可8色距离2着色）的证明，重点简化了原证明中移除5-环这一技术难点部分。
The sharp upper bounds on the maximum degree and vertex-connectivity of claw-free 1-planar graphs
本文研究无爪1-平面图，证明其最大度上界为10且尖锐，在禁止诱导爪时6-连通和最优1-平面图最大度上界为8，并证明7-连通1-平面图必含诱导爪，从而顶点连通度至多6。
Let $f(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. The order of magnitude of $f(n,P_k)$, where $P_k$ is a path on $k$ vertices, is $n^{{\lfloor{\frac{k-1}{2}}\rfloor}+1}$. In this paper we determine the asymptotic value of $f(n,P_5)$ and give conjectures for longer paths.
Frugal coloring in planar graphs with maximum degree 4 and 5
该研究针对最大度为4和5的平面图，证明其2-节俭着色分别可使用7种和10种颜色，是Wegner猜想的部分变体结果。
Let P_{n,d,D} denote the graph taken uniformly at random from the set of all labelled planar graphs on {1,2,...,n} with minimum degree at least d(n) and maximum degree at most D(n). We use counting arguments to investigate the probability that P_{n,d,D} wll contain given components and subgraphs, showing exactly when this is bounded away from 0 and 1 as n tends to infinity.
Akiyama and Watanabe conjectured that every simple planar bipartite graph on $n$ vertices contains an induced forest on at least $5n/8$ vertices. We apply the discharging method to show that every simple bipartite planar graph on $n$ vertices contains an induced forest on at least $\lceil (4n+3)/7 \rceil$ vertices.
A total $k$-coloring of a graph is an assignment of $k$ colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The Total Coloring Conjecture (TCC) states that every simple graph $G$ has a total ($Δ(G)+2$)-coloring, where $Δ(G)$ is the maximum degree of $G$. This conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6. It is known that every planar graph $G$ of $Δ(G) \geq 9$ or $Δ(G) \in \{7, 8\}$ with some restrictions has a total $(Δ(G) + 1)$-colori…
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…
The maximum number of $k$-cliques of 7-connected 1-planar graphs
研究7连通1平面图的团数上限，证明其边数、三角形数和K4数量均有限制，总团数不超过10n-33，所有界在无穷多个n值下最优。
Total-coloring of planar graphs with maximum degree 6 and without prescribed 4-cycles
本文改进了平面图全染色结果，证明最大度为6且不含某些特殊4环的平面图是7-全可染的，推进了全染色猜想在平面图上的研究。
Let $f(n,H)$ denote the maximum number of copies of $H$ possible in an $n$-vertex planar graph. The function $f(n,H)$ has been determined when $H$ is a cycle of length $3$ or $4$ by Hakimi and Schmeichel and when $H$ is a complete bipartite graph with smaller part of size 1 or 2 by Alon and Caro. We determine $f(n,H)$ exactly in the case when $H$ is a path of length 3.
The spread of a graph $G$ is the difference between the largest and smallest eigenvalue of the adjacency matrix of $G$. Gotshall, O'Brien and Tait conjectured that for sufficiently large $n$, the $n$-vertex outerplanar graph with maximum spread is the graph obtained by joining a vertex to a path on $n-1$ vertices. In this paper, we disprove this conjecture by showing that the extremal graph is the graph obtained by joining a vertex to a path on $\lceil (2n-1)/3\rceil$ vertices and $\lfloor(n-2)/3\rfloor$ isolated vertices. For planar graphs, we show that the extremal $n$-vertex planar graph at…
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 $6\leqslant Δ\leqslant 31$.
Equitable coloring of planar graphs with maximum degree at least eight
本文证明了Chen-Lih-Wu猜想对最大度至少为8的平面图及最大度至少为9的可嵌入非负欧拉示性数曲面的图成立，扩展了已知结果。
On 2-distance 16-coloring of planar graphs with maximum degree at most five
针对最大度不超过5的平面图的2-距离染色问题，证明其可进行2-距离16染色，改进了Hou等人2023年的相关研究结果。
Induced planar Turán numbers
本文研究了禁止诱导子图的平面图Turán数，确定了当F为四边形加对角线时的上界，并在F为路径图时得到精确极值。
In the past various distance based colorings on planar graphs were introduced. We turn our focus to three of them, namely $2$-distance coloring, injective coloring, and exact square coloring. A $2$-distance coloring is a proper coloring of the vertices in which no two vertices at distance $2$ receive the same color, an injective coloring is a coloring of the vertices in which no two vertices with a common neighbor receive the same color, and an exact square coloring is a coloring of the vertices in which no two vertices at distance exactly $2$ receive the same color. We prove that planar graph…
For a graph $G$, by $χ_2(G)$ we denote the minimum integer $k$, such that there is a $k$-coloring of the vertices of $G$ in which vertices at distance at most 2 receive distinct colors. Equivalently, $χ_2(G)$ is the chromatic number of the square of $G$. In 1977 Wegner conjectured that if $G$ is planar and has maximum degree $Δ$, then $χ_2(G) \leq 7$ if $Δ\leq 3$, $χ_2(G) \leq Δ+5$ if $4 \leq Δ\leq 7$, and $\lfloor 3Δ/2 \rfloor +1$ if $Δ\geq 8$. Despite extensive work, the known upper bounds are quite far from the conjectured ones, especially for small values of $Δ$. In this work we show that …
A graph $G$ is $d$-degenerate if every non-null subgraph of $G$ has a vertex of degree at most $d$. We prove that every $n$-vertex planar graph has a $3$-degenerate induced subgraph of order at least $3n/4$.
For a fixed 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 the case when $H$ is a cycle, the asymptotic value of $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,C_m)$ is currently known for $m\in\{3,4,5,6,8\}$. In this note, we extend this list by establishing $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,C_{10})\sim(n/5)^5$ and $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,C_{12})\sim(n/6)^6$. We prove this by answering the following question for $m\in\{5,6\}$, which is interesting in its own right: which …
A sufficient condition for planar graphs with maximum degree eight to be totally 9-colorable
针对最大度为8的平面图的全着色问题，证明了不含4-扇子图的此类图全色数为9，给出了平面图全9-可着色的一个充分条件。
2-distance 20-coloring of planar graphs with maximum degree 6
本文证明最大度至多6的平面图具有2-距离20染色，将已知上界从21改进至20。
A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this note we give examples of class two 1-planar graphs with maximum degree six or seven.
Improved 2-Distance Coloring of Planar Graphs with Maximum Degree 5
本文研究了平面图的2-距离染色问题，证明了最大度不超过5的平面图的2-距离色数上界可由18改进至17。
We show that the size of maximum cut in a planar graph with $m$ edges is at least $2m/3$. We also show that maximal planar graphs saturate this bound.
Planar graphs with the maximum number of induced 6-cycles
本文确定了大 $n$ 时 $n$ 顶点平面图中诱导6-环的最大数量，刻画了极值构造，并将作者此前关于4-环、5-环的结果推广到6-环。
The maximum matching extendability and factor-criticality of 1-planar graphs
本文研究了1-平面图的匹配扩展性和因子临界性，证明了最优1-平面图无法3-可扩展，且无1-平面图可5-可扩展，同时构造了4-可扩展的1-平面图，并指出无图可7-因子临界。
In a generalized Turán problem, we are given graphs $H$ and $F$ and seek to maximize the number of copies of $H$ in an $F$-free graph of order $n$. We consider generalized Turán problems where the host graph is planar. In particular we obtain the order of magnitude of the maximum number of copies of a fixed tree in a planar graph containing no even cycle of length at most $2\ell$, for all $\ell$, $\ell \geq 1$. We obtain the order of magnitude of the maximum number of cycles of a given length in a planar $C_4$-free graph. An exact result is given for the maximum number of $5$-cycles in a $C_4$…
Square Coloring of Planar Graphs with Maximum Degree at Most Five
本文研究了最大度至多为5的平面图的平方染色问题，证明了其平方图可用至多17种颜色正常顶点染色，改进了此前18种颜色的上界。
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 maximum number of odd cycles in a planar graph
本文研究了平面图中固定奇圈的最大数量问题，针对特定$m$值给出了渐近解答，并证明了其他$m$值在$3/2$因子内紧致的界，将问题归约为团边上的最大似然概率问题。
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…
The ($d$,1)-total labelling of graphs was introduced by Havet and Yu. In this paper, we prove that, for planar graph $G$ with maximum degree $Δ\geq12$ and $d=2$, the (2,1)-total labelling number $λ_2^T(G)$ is at most $Δ+2$.
Between proper and square colorings of planar graphs with maximum degree at most four
本文研究平面图最大度数不超过四的着色问题，证明了该类图可被(1,2^10)和(1^2,2^7)着色，为Wegner猜想提供新进展。
Relaxation of Wegner's Planar Graph Conjecture for maximum degree 4
针对平面图最大度数为4的情况，研究提出了一种新的着色方法，证明使用9种颜色即可正确着色其平方图。
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.
Partitioning triangle-free planar graphs into a forest and a linear forest
本文改进了无三角形平面图的划分结果，证明其可划分为森林和线性森林。
The maximum number of paths of even length in a planar graph
该研究证明了Ghosh等人关于平面图中偶长度路径最大数量的猜想，同时解决了Cox–Martin优化猜想。
This paper considers the conjecture by Grünbaum that every planar 3-connected graph has a spanning tree $T$ such that both $T$ and its co-tree have maximum degree at most 3. Here, the co-tree of $T$ is the spanning tree of the dual obtained by taking the duals of the non-tree edges. While Grünbaum's conjecture remains open, we show that every planar 3-connected graph has a spanning tree $T$ such that both $T$ and its co-tree have maximum degree at most 5. It can be found in linear time.
Here we prove that counting maximum matchings in planar, bipartite graphs is #P-complete. This is somewhat surprising in the light that the number of perfect matchings in planar graphs can be computed in polynomial time. We also prove that counting non-necessarily perfect matchings in planar graphs is already #P-complete if the problem is restricted to bipartite graphs. So far hardness was proved only for general, non-necessarily bipartite graphs.
We consider the problem of list edge coloring for planar graphs. Edge coloring is the problem of coloring the edges while ensuring that two edges that are incident receive different colors. A graph is k-edge-choosable if for any assignment of k colors to every edge, there is an edge coloring such that the color of every edge belongs to its color assignment. Vizing conjectured in 1965 that every graph is (D+1)-edge-choosable, where D is the maximum degree. In 1990, Borodin solved the conjecture for planar graphs with maximum degree at least 9, and asked whether the bound could be lowered to 8. …
We show that the left/right relation on the set of s-t-paths of a plane graph induces a so-called submodular lattice. If the embedding of the graph is s-t-planar, this lattice is even consecutive. This implies that Ford and Fulkerson's uppermost path algorithm for maximum flow in such graphs is indeed a special case of a two-phase greedy algorithm on lattice polyhedra. We also show that the properties submodularity and consecutivity cannot be achieved simultaneously by any partial order on the paths if the graph is planar but not s-t-planar, thus providing a characterization of this class of g…
The maximum number of edges of bipartite 1-planar graphs with 1-disk drawings
本文解决了具有1-圆盘OX绘图的二分1-平面图的最大边数问题，证明其上界为2|V(G)|+|X|-6，并证明该界是紧的。
A graph is called dispersable if it has a book embedding in which each page has maximum degree 1 and the number of pages is the maximum degree. Bernhart and Kainen conjectured every k-regular bipartite graph is dispersable. Forty years later, Alam, Bekos, Gronemann, Kaufmann, and Pupyrev have disproved this conjecture, identifying nonplanar 3- and 4-regular bipartite graphs that are not dispersable. They also proved all cubic planar bipartite 3-connected graphs are dispersable and conjectured that the connectivity condition could be relaxed. We prove that every cubic planar bipartite multigrap…
The maximum number of two 4-vertex graphs in planar graphs
研究平面图中两个四顶点图K_{1,3}^+和K_4^-的最大可能数目，通过构造和极值分析得到精确公式。
We study the \emph{maximum differential coloring problem}, where the vertices of an $n$-vertex graph must be labeled with distinct numbers ranging from $1$ to $n$, so that the minimum absolute difference between two labels of any two adjacent vertices is maximized. As the problem is \NPH for general graphs~\cite{leung1984}, we consider planar graphs and subclasses thereof. We initially prove that the maximum differential coloring problem remains \NPH, even for planar graphs. Then, we present tight bounds for regular caterpillars and spider graphs. Using these new bounds, we prove that the Mill…
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