Yong Yu, Xin Zhang, Guanghui Wang, Jinbo Li
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 $Δ+2$.
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Laman graphs naturally arise in structural mechanics and rigidity theory. Specifically, they characterize minimally rigid planar bar-and-joint systems which are frequently needed in robotics, as well as in molecular chemistry and polymer physics. We introduce three new combinatorial structures for planar Laman graphs: angular structures, angle labelings, and edge labelings. The latter two structures are related to Schnyder realizers for maximally planar graphs. We prove that planar Laman graphs are exactly the class of graphs that have an angular structure that is a tree, called angular tree, …
Approximate proof labeling schemes were introduced by \\Censor-Hillel, Paz and Perry \cite{CPP}. Roughly speaking, a graph property~$\cP$ can be verified by an approximate proof labeling scheme in constant-time if the vertices of a graph having the property can be convinced, in a short period of time not depending on the size of the graph, that they are having the property $\cP$ or at least they are not far from being having the property $\cP$. The main result of this paper is that bounded-degree planar graphs (and also outer-planar graphs, bounded genus graphs, knotlessly embeddable graphs et…
We investigate adjacency labeling schemes for graphs of bounded degree $Δ= O(1)$. In particular, we present an optimal (up to an additive constant) $\log n + O(1)$ adjacency labeling scheme for bounded degree trees. The latter scheme is derived from a labeling scheme for bounded degree outerplanar graphs. Our results complement a similar bound recently obtained for bounded depth trees [Fraigniaud and Korman, SODA 10], and may provide new insights for closing the long standing gap for adjacency in trees [Alstrup and Rauhe, FOCS 02]. We also provide improved labeling schemes for bounded degree p…
A vertex k-labeling of graph G is distinguishing if the only automorphism that preserves the labels of G is the identity map. The distinguishing number of G, D(G), is the smallest integer k for which G has a distinguishing k-labeling. In this paper, we apply the principle of inclusion-exclusion and develop recursive formulas to count the number of inequivalent distinguishing k-labelings of a graph. Along the way, we prove that the distinguishing number of a planar graph can be computed in time polynomial in the size of the graph.}
On homomorphism related parameters of oriented triangle-free planar graphs
研究无向三角形平面图的同态相关参数，证明其定向相对团数为10，还证明特定条件下基础图最小度至少为10，借此得出一些参数的改进下界。
An L(2, 1)-labeling of a graph is an assignment of nonnegative integers to the vertices of G such that adjacent vertices receive numbers differed by at least 2, and vertices at distance 2 are assigned distinct numbers. The L(2, 1)-labeling number is the minimum range of labels over all such labeling. It was shown by Griggs and Yeh [Labelling graphs with a condition at distance 2, SIAM J. Discrete Math. 5(1992), 586-595] that the L(2, 1)-labeling number of a tree is either \D+ 1 or \D + 2. In this paper, we give a complete characterization of L(2, 1)-labeling number for trees with maximum degre…
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.
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.
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.
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. …
Maximizing the number of integer pairs summing to powers of 2 via graph labeling and solving restricted systems of linear (in)equations
通过图标记和求解受限线性方程组，研究如何最大化整数对求和为2的幂次的数量，并确定最大图结构及禁止子图。
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…
We extend a result of Griggs and Yeh about the maximum possible value of the L(2,1)-labeling number of a graph in terms of its maximum degree to oriented graphs. We consider the problem both in the usual definition of the oriented L(2,1)-labeling number and in some variants we introduce.
We give bounds on the L(2,1)-labeling number of a simple graph in terms of its order and its maximum degree. We also describe an infinite class of graphs of which the elements have the highest L(2,1)-labeling numbers in terms of their maximum degrees of any known infinite class of graphs.
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…
Refining a classical proof of Whitney, we show that any $4$-connected planar triangulation can be decomposed into a Hamiltonian path and two trees. Therefore, every $4$-connected planar graph decomposes into three forests, one having maximum degree at most $2$. We use this result to show that any Hamiltonian planar triangulation can be decomposed into two trees and one spanning tree of maximum degree at most $3$. These decompositions improve the result of Gonçalves [Covering planar graphs with forests, one having bounded maximum degree. J. Comb. Theory, Ser. B, 100(6):729--739, 2010] that ever…
An $L(2, 1)$-labeling of a graph $G$ is an assignment of a nonnegative integer to each vertex of $G$ such that adjacent vertices receive integers that differ by at least two and vertices at distance two receive distinct integers. The span of such a labeling is the difference between the largest and smallest integers used. The $λ$-number of $G$, denoted by $λ(G)$, is the minimum span over all $L(2, 1)$-labelings of $G$. Bodlaender {\it et al.} conjectured that if $G$ is an outerplanar graph of maximum degree $Δ$, then $λ(G)\leq Δ+2$. Calamoneri and Petreschi proved that this conjecture is true …
A graph $G$ is $k$-degenerate if it can be transformed into an empty graph by subsequent removals of vertices of degree $k$ or less. We prove that every connected planar graph with average degree $d \ge 2$ has a 4-degenerate induced subgraph containing at least $(38-d)/36$ of its vertices. This shows that every planar graph of order $n$ has a 4-degenerate induced subgraph of order more than $8/9 \cdot n$. We also consider a local variation of this problem and show that in every planar graph with at least 7 vertices, deleting a suitable vertex allows us to subsequently remove at least 6 more ve…
Largest planar graphs of diameter $3$ and fixed maximum degree -- connection with fractional matchings
针对直径3、最大度固定的平面图，本文借助分数最大匹配证明最大顶点数上界为9/2Δ+c，并刻画极值图。
A {\em conflict-free coloring} of a graph {\em with respect to open} (resp., {\em closed}) {\em neighborhood} is a coloring of vertices such that for every vertex there is a color appearing exactly once in its open (resp., closed) neighborhood. Similarly, a {\em unique-maximum coloring} of a graph {\em with respect to open} (resp., {\em closed}) {\em neighborhood} is a coloring of vertices such that for every vertex the maximum color appearing in its open (resp., closed) neighborhood appears exactly once. There is a vast amount of literature on both notions where the colorings need not be prop…
Minimum cuts have been closely related to shortest paths in planar graphs via planar duality - so long as the graphs are undirected. Even maximum flows are closely related to shortest paths for the same reason - so long as the source and the sink are on a common face. In this paper, we give a correspondence between maximum flows and shortest paths via duality in directed planar graphs with no constraints on the source and sink. We believe this a promising avenue for developing algorithms that are more practical than the current asymptotically best algorithms for maximum st-flow.
We determine the maximum number of edges that a planar graph can have as a function of its maximum degree and matching number.
On the maximum number of edges of outer k-planar graphs
研究平面上凸位置点集在每条线段最多与$k$条其他线段相交条件下的最大连接数，即外$k$-平面图的最大边数，得出上界为$(\sqrt{2}+\varepsilon)\sqrt{k}n$，并扩展探讨了二分图及循环图最大割问题。
A graph is $(k_1,k_2)$-colorable if its vertex set can be partitioned into a graph with maximum degree at most $k_1$ and and a graph with maximum degree at most $k_2$. We show that every $(C_3,C_4,C_6)$-free planar graph is $(0,6)$-colorable. We also show that deciding whether a $(C_3,C_4,C_6)$-free planar graph is $(0,3)$-colorable is NP-complete.
We consider the problem of coloring the squares of graphs of bounded maximum average degree, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbour receive different colors. Borodin et al. proved in 2004 and 2008 that the squares of planar graphs of girth at least seven and sufficiently large maximum degree $Δ$ are list $(Δ+1)$-colorable, while the squares of some planar graphs of girth six and arbitrarily large maximum degree are not. By Euler's Formula, planar graphs of girth at least $6$ are of maximum average degree less …
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.
A graph is outer-1-planar if it can be drawn in the plane so that all vertices are on the outer face and each edge is crossed at most once. It is known that the list edge chromatic number $χ'_l(G)$ of any outer-1-planar graph $G$ with maximum degree $Δ(G)\geq 5$ is exactly its maximum degree. In this paper, we prove $χ'_l(G)=Δ(G)$ for outer-1-planar graphs $G$ with $Δ(G)=4$ and with the crossing distance being at least 3.
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 $n$.
We show that the edges of any planar graph of maximum degree at most $9$ can be partitioned into $4$ linear forests and a matching. Combined with known results, this implies that the edges of any planar graph $G$ of odd maximum degree $Δ\ge 9$ can be partitioned into $\tfrac{Δ-1}{2}$ linear forests and one matching. This strengthens well-known results stating that graphs in this class have chromatic index $Δ$ [Vizing, 1965] and linear arboricity at most $\lceil(Δ+1)/2\rceil$ [Wu, 1999].
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.
The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has a vertex (edge) labeling with $d$ labels that is preserved only by a trivial automorphism. In this paper we consider the maximal outerplanar graphs (MOP graphs) and show that MOP graphs, except $K_3$, can be distinguished by at most two vertex (edge) labels. We also compute the distinguishing number and the distinguishing index of Halin and Mycielskian graphs.
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.
Quasiperiods of Magic Labeling Quasipolynomials
该研究采用Ehrhart理论按最大边标号统计图的幻标号，证明对应拟多项式的最小拟周期可任意大，排除了用该类方法证明反幻图猜想的可能，同时证明匹配毁数≤1的二部图对应的该函数是多项式。
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.
The bend-number b(G) of a graph G is the minimum k such that G may be represented as the edge intersection graph of a set of grid paths with at most k bends. We confirm a conjecture of Biedl and Stern showing that the maximum bend-number of outerplanar graphs is 2. Moreover we improve the formerly known lower and upper bound for the maximum bend-number of planar graphs from 2 and 5 to 3 and 4, respectively.
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 $C_5$…
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 paper, we confirm the total-coloring conjecture for 1-planar graphs with maximum degree at least 13.
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。
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.
The Fiedler value $λ_2$, also known as algebraic connectivity, is the second smallest Laplacian eigenvalue of a graph. We study the maximum Fiedler value among all planar graphs $G$ with $n$ vertices, denoted by $λ_{2\max}$, and we show the bounds $2+Θ(\frac{1}{n^2}) \leq λ_{2\max} \leq 2+O(\frac{1}{n})$. We also provide bounds on the maximum Fiedler value for the following classes of planar graphs: Bipartite planar graphs, bipartite planar graphs with minimum vertex degree~3, and outerplanar graphs. Furthermore, we derive almost tight bounds on $λ_{2\max}$ for two more classes of graphs, thos…
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.
We prove the existence of a function $f :\mathbb{N} \to \mathbb{N}$ such that the vertices of every planar graph with maximum degree $Δ$ can be 3-colored in such a way that each monochromatic component has at most $f(Δ)$ vertices. This is best possible (the number of colors cannot be reduced and the dependence on the maximum degree cannot be avoided) and answers a question raised by Kleinberg, Motwani, Raghavan, and Venkatasubramanian in 1997. Our result extends to graphs of bounded genus.
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.
We settle a problem of Dujmović, Eppstein, Suderman, and Wood by showing that there exists a function $f$ with the property that every planar graph $G$ with maximum degree $d$ admits a drawing with noncrossing straight-line edges, using at most $f(d)$ different slopes. If we allow the edges to be represented by polygonal paths with {\em one} bend, then 2d slopes suffice. Allowing {\em two} bends per edge, every planar graph with maximum degree $d\ge 3$ can be drawn using segments of at most $\lceil d/2\rceil$ different slopes. There is only one exception: the graph formed by the edges of an oc…
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…
We prove that planar graphs of maximum degree 3 and of girth at least 7 are 3-edge-colorable, extending the previous result for girth at least 8 by Kronk, Radlowski, and Franen from 1974.
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…
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。
Boolean dimension and dim-boundedness: Planar cover graph with a zero
本文针对带唯一极小元的平面覆盖图偏序集，证明其布尔维数至多为10，进而推导此类平面有向图存在O(log n)位的可达性标记方案，同时补充了相关维数理论的研究结果。
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。
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