François Dross
We show that every 1-planar graph with minimum degree at least 4 has girth at most $8$, and every 1-planar graph with minimum degree at least 3 has girth at most $198$.
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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.
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 $\frac{44n+50}{69}$.
Recently, Borodin, Kostochka, and Yancey (On $1$-improper $2$-coloring of sparse graphs. Discrete Mathematics, 313(22), 2013) showed that the vertices of each planar graph of girth at least $7$ can be $2$-colored so that each color class induces a subgraph of a matching. We prove that any planar graph of girth at least $6$ admits a vertex coloring in $2$ colors such that each monochromatic component is a path of length at most $14$. Moreover, we show a list version of this result. On the other hand, for each positive integer $t\geq 3$, we construct a planar graph of girth $4$ such that in any …
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 $\frac{n+2}{3}$.
2-distance 4-coloring of planar subcubic graphs with girth at least 21
本文证明了围长至少为21的平面次立方图具有2-距离4-染色，并构造出围长为11的平面次立方图反例，界定了该染色性质成立的围长下界。
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 has girth at least six and all lists have size at least three, then there exists an L-coloring respecting at least a constant fraction of the preferences.
We denote by $χ$ g (G) the game chromatic number of a graph G, which is the smallest number of colors Alice needs to win the coloring game on G. We know from Montassier et al. [M. Montassier, P. Ossona de Mendez, A. Raspaud and X. Zhu, Decomposing a graph into forests, J. Graph Theory Ser. B, 102(1):38-52, 2012] and, independantly, from Wang and Zhang, [Y. Wang and Q. Zhang. Decomposing a planar graph with girth at least 8 into a forest and a matching, Discrete Maths, 311:844-849, 2011] that planar graphs with girth at least 8 have game chromatic number at most 5. One can ask if this bound of …
Given a graph $G$ and a list assignment $L(v)$ for each vertex of $v$ of $G$. A proper $L$-list-coloring of $G$ is a function that maps every vertex to a color in $L(v)$ such that no pair of adjacent vertices have the same color. We say that a graph is list $k$-colorable when every vertex $v$ has a list of colors of size at least $k$. A $2$-distance coloring is a coloring where vertices at distance at most 2 cannot share the same color. We prove the existence of a $2$-distance list ($Δ+2$)-coloring for planar graphs with girth at least $10$ and maximum degree $Δ\geq 4$.
Partial cubes are graphs isometrically embeddable into hypercubes. We analyze how isometric cycles in partial cubes behave and derive that every partial cube of girth more than 6 must have vertices of degree less than 3. As a direct corollary we get that every regular partial cube of girth more than 6 is an even cycle. Along the way we prove that every partial cube $G$ with girth more than 6 is a tree-zone graph and therefore $2n(G)-m(G)-i(G)+ce(G)=2$ holds, where $i(G)$ is the isometric dimension of $G$ and $ce(G)$ its convex excess.
Given a multigraph $G$ and a positive integer $t$, the distance-$t$ chromatic index of $G$ is the least number of colours needed for a colouring of the edges so that every pair of distinct edges connected by a path of fewer than $t$ edges must receive different colours. Let $π'_t(d)$ and $τ'_t(d)$ be the largest values of this parameter over the class of planar multigraphs and of (simple) trees, respectively, of maximum degree $d$. We have that $π'_t(d)$ is at most and at least a non-trivial constant multiple larger than $τ'_t(d)$. (We conjecture $\limsup_{d\to\infty}π'_2(d)/τ'_2(d) =9/4$ in p…
String Graph Obstacles of High Girth and of Bounded Degree
本文研究弦图的极小非弦图障碍，构造围长四、无$K_{2,3}$子图的近平面无穷障碍族，证明次三次障碍的围长界限，并给出有界树宽情形的线性时间识别算法。
The girth of a graph is the length of its shortest cycle. We give an algorithm that computes in O(n(log n)^3) time and O(n) space the (weighted) girth of an n-vertex planar digraph with arbitrary real edge weights. This is an improvement of a previous time bound of O(n^(3/2)), a bound which was only valid for non-negative edge-weights. Our algorithm can be modified to output a shortest cycle within the same time and space bounds if such a cycle exists.
As an extension of the Four-Color Theorem it is conjectured that every planar graph of odd-girth at least $2k+1$ admits a homomorphism to $PC_{2k}=(\mathbb{Z}_2^{2k}, \{e_1, e_2, ...,e_{2k}, J\})$ where $e_i$'s are standard basis and $J$ is all 1 vector. Noting that $PC_{2k}$ itself is of odd-girth $2k+1$, in this work we show that if the conjecture is true, then $PC_{2k}$ is an optimal such a graph both with respect to number of vertices and number of edges. The result is obtained using the notion of walk-power of graphs and their clique numbers. An analogous result is proved for bipartite si…
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 …
A half-square of a bipartite graph $B=(X,Y,E_B)$ has one color class of $B$ as vertex set, say $X$; two vertices are adjacent whenever they have a common neighbor in $Y$. If $G=(V,E_G)$ is the half-square of a planar bipartite graph $B=(V,W,E_B)$, then $G$ is called a map graph, and $B$ is a witness of $G$. Map graphs generalize planar graphs, and have been introduced and investigated by Chen, Grigni and Papadimitriou [STOC 1998, J. ACM 2002]. They proved that recognizing map graphs is in $\mathsf{NP}$ by proving the existence of a witness. Soon later, Thorup [FOCS 1998] claimed that recognizi…
A strong edge-colouring of a graph is a proper edge-colouring where each colour class induces a matching. It is known that every planar graph with maximum degree $Δ$ has a strong edge-colouring with at most $4Δ+4$ colours. We show that $3Δ+1$ colours suffice if the graph has girth 6, and $4Δ$ colours suffice if $Δ\geq 7$ or the girth is at least 5. In the last part of the paper, we raise some questions related to a long-standing conjecture of Vizing on proper edge-colouring of planar graphs.
We investigate the zero-forcing number for triangle-free graphs. We improve upon the trivial bound, $δ\le Z(G)$ where $δ$ is the minimum degree, in the triangle-free case. In particular, we show that $2 δ- 2 \le Z(G)$ for graphs with girth of at least 5, and this can be further improved when $G$ has a small cut set. Using these results, we are able to prove the Graph Complement Conjecture on minimum rank for a large class of graphs. Lastly, we make a conjecture that the lower bound for $Z(G)$ increases as a function of the girth, $g$, and $δ$.
We present unified bijections for maps on the torus with control on the face-degrees and essential girth (girth of the periodic planar representation). A first step is to show that for d>=3 every toroidal d-angulation of essential girth d can be endowed with a certain "canonical" orientation (formulated as a weight-assignment on the half-edges). Using an adaptation of a construction by Bernardi and Chapuy, we can then derive a bijection between face-rooted toroidal d-angulations of essential girth d (with the condition that, apart from the root-face contour, no other closed walk of length d en…
Locating-dominating sets and identifying codes are two closely related notions in the area of separating systems. Roughly speaking, they consist in a dominating set of a graph such that every vertex is uniquely identified by its neighbourhood within the dominating set. In this paper, we study the size of a smallest locating-dominating set or identifying code for graphs of girth at least 5 and of given minimum degree. We use the technique of vertex-disjoint paths to provide upper bounds on the minimum size of such sets, and construct graphs who come close to meet these bounds.
We establish a lower bound for the cop number of graphs of high girth in terms of the minimum degree, and more generally, in terms of a certain growth condition. We show, in particular, that the cop number of any graph with girth $g$ and minimum degree $δ$ is at least $\tfrac{1}{g}(δ- 1)^{\lfloor \frac{g-1}{4}\rfloor}$. We establish similar results for directed graphs. While exposing several reasons for conjecturing that the exponent $\tfrac{1}{4}g$ in this lower bound cannot be improved to $(\tfrac{1}{4}+\varepsilon)g$, we are also able to prove that it cannot be increased beyond $\frac{3}{8}…
A catalog of a class of (3,g) graphs for even girth g is introduced in this paper. A (k,g) graph is a regular graph with degree k and girth g. This catalog of (3,g) graphs for even girth g satisfying 6 <= g <= 16, has the following properties. Firstly, this catalog contains the smallest known (3, g) graphs. An appropriate class of cubic graphs for this catalog has been identified, such that the (3,g) graph of minimum order within the class is also the smallest known (3,g) graph. Secondly, this catalog contains (3,g) graphs for more orders than other listings. Thirdly, the class of graphs have …
Using computational techniques we provide a framework for proving results on subclasses of planar graphs via discharging method. The aim of this paper is to apply these techniques to study the 2-distance coloring of planar subcubic graphs. Applying these techniques we show that every subcubic planar graph $G$ of girth at least 8 has 2-distance chromatic number at most 6.
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…
In this paper, we aim to introduce the group version of edge coloring and list edge coloring, and prove that all 2-degenerate graphs along with some planar graphs without adjacent short cycles is group $(Δ(G)+1)$-edge-choosable while some planar graphs with large girth and maximum degree is group $Δ(G)$-edge-choosable.
We study a variant of intersection representations with unit balls, that is, unit disks in the plane and unit intervals on the line. Given a planar graph and a bipartition of the edges of the graph into near and far sets, the goal is to represent the vertices of the graph by unit balls so that the balls representing two adjacent vertices intersect if and only if the corresponding edge is near. We consider the problem in the plane and prove that it is NP-hard to decide whether such a representation exists for a given edge-partition. On the other hand, every series-parallel graph admits such a r…
In this paper, we obtain lower bounds for the domination numbers of connected graphs with girth at least $7$. We show that the domination number of a connected graph with girth at least $7$ is either $1$ or at least $\frac{1}{2}(3+\sqrt{8(m-n)+9})$, where $n$ is the number of vertices in the graph and $m$ is the number of edges in the graph. For graphs with minimum degree $2$ and girth at least $7$, the lower bound can be improved to $\max{\{\sqrt{n}, \sqrt{\frac{2m}{3}}\}}$, where $n$ and $m$ are the numbers of vertices and edges in the graph respectively. In cases where the graph is of minim…
Isometric Cycles and a Generalization of Moore Graphs
该研究通过图的赤道长度、围长和最小度给出图阶数的下界，定义并刻画了达到下界的类Moore极图的结构性质，还推导了无C4图的相关阶数界并提出相关猜想。
A graph $G$ is hypohamiltonian if $G$ is non-hamiltonian and $G - v$ is hamiltonian for every $v \in V(G)$. In the following, every graph is assumed to be hypohamiltonian. Aldred, Wormald, and McKay gave a list of all graphs of order at most 17. In this article, we present an algorithm to generate all graphs of a given order and apply it to prove that there exist exactly 14 graphs of order 18 and 34 graphs of order 19. We also extend their results in the cubic case. Furthermore, we show that (i) the smallest graph of girth 6 has order 25, (ii) the smallest planar graph has order at least 23, (…
We describe an algorithm for generating all $k$-critical $\mathcal H$-free graphs, based on a method of Hoàng et al. Using this algorithm, we prove that there are only finitely many $4$-critical $(P_7,C_k)$-free graphs, for both $k=4$ and $k=5$. We also show that there are only finitely many $4$-critical graphs $(P_8,C_4)$-free graphs. For each case of these cases we also give the complete lists of critical graphs and vertex-critical graphs. These results generalize previous work by Hell and Huang, and yield certifying algorithms for the $3$-colorability problem in the respective classes. More…
Signed graphs are studied since the middle of the last century. Recently, the notion of homomorphism of signed graphs has been introduced since this notion captures a number of well known conjectures which can be reformulated using the definitions of signed homomorphism. In this paper, we introduce and study the properties of some target graphs for signed homomorphism. Using these properties, we obtain upper bounds on the signed chromatic numbers of graphs with bounded acyclic chromatic number and of signed planar graphs with given girth.
A set $S\subseteq V(G)$ of a graph $G$ is a dominating set if each vertex has a neighbor in $S$ or belongs to $S$. Let $γ(G)$ be the cardinality of a minimum dominating set in $G$. The bondage number $b(G)$ of a graph $G$ is the smallest cardinality of a set of edges $A\subseteq E(G)$, such that $γ(G-A)=γ(G)+1$. The $d$-Bondage is the problem of deciding, given a graph $G$ and an integer $d\geq 1$, if $b(G)\leq d$. This problem is known to be $\mathsf{NP}$-hard even for bipartite graphs and $d=1$. In this paper, we show that $1$-Bondage is $\mathsf{NP}$-hard, even for the class of $3$-regular …
Asymmetric Homomorphism Thresholds for Graphs of Large Odd Girth
本文确定了奇数围长至少为7的图到无三角形图的不对称同态阈值为1/9，并推广到更一般的K_t及任意奇数围长情形，否证了相关猜想，下界基于高维Borsuk图，上界使用正则性方法。
An injective $k$-edge-coloring of a graph $G$ is an assignment of colors, i.e. integers in $\{1, \ldots , k\}$, to the edges of $G$ such that any two edges each incident with one distinct endpoint of a third edge, receive distinct colors. The problem of determining whether such a $k$-coloring exists is called k-INJECTIVE EDGE-COLORING. We show that 3-INJECTIVE EDGE-COLORING is NP-complete, even for triangle-free cubic graphs, planar subcubic graphs of arbitrarily large girth, and planar bipartite subcubic graphs of girth~6. 4-INJECTIVE EDGE-COLORING remains NP-complete for cubic graphs. For an…
Smallest Cubic Non-1-Planar Graphs
本文证明最小的三次非1-平面图有30个顶点，并借助SAT求解器与k-灵活性方法，避免了穷举四十亿个图的计算。
Proper conflict-free choosability of planar graphs
本文证明了特定平面图类的适当无冲突可选性，包括最小圈长至少12的平面图和外1-平面图，且证明了这些图类的适当无冲突6-可选性。
Every triangle-free planar graph on n vertices has an independent set of size at least (n+1)/3, and this lower bound is tight. We give an algorithm that, given a triangle-free planar graph G on n vertices and an integer k>=0, decides whether G has an independent set of size at least (n+k)/3, in time 2^{O(sqrt{k})}n. Thus, the problem is fixed-parameter tractable when parameterized by k. Furthermore, as a corollary of the result used to prove the correctness of the algorithm, we show that there exists epsilon>0 such that every planar graph of girth at least five on n vertices has an independent…
A $k$-subcoloring of a graph is a partition of the vertex set into at most $k$ cluster graphs, that is, graphs with no induced $P_3$. 2-subcoloring is known to be NP-complete for comparability graphs and three subclasses of planar graphs, namely triangle-free planar graphs with maximum degree 4, planar perfect graphs with maximum degree 4, and planar graphs with girth 5. We show that 2-subcoloring is also NP-complete for planar comparability graphs with maximum degree 4.
The Additive Coloring Problem is a variation of the Coloring Problem where labels of $\{1,\ldots,k\}$ are assigned to the vertices of a graph $G$ so that the sum of labels over the neighborhood of each vertex is a proper coloring of $G$. The least value $k$ for which $G$ admits such labeling is called \emph{additive chromatic number} of $G$. This problem was first presented by Czerwiński, Grytczuk and Żelazny who also proposed a conjecture that for every graph $G$, the additive chromatic number never exceeds the classic chromatic number. Up to date, the conjecture has been proved for complete …
It is well-known that 1-planar graphs have minimum degree at most 7, and not hard to see that some 1-planar graphs have minimum degree exactly 7. In this note we show that any such 1-planar graph has at least 24 vertices, and this is tight.
An equitable $(t,k,d)$-tree-coloring of a graph $G$ is a coloring to vertices of $G$ such that the sizes of any two color classes differ by at most one and the subgraph induced by each color class is a forest of maximum degree at most $k$ and diameter at most $d$. The minimum $t$ such that $G$ has an equitable $(t',k,d)$-tree-coloring for every $t'\geq t$ is called the strong equitable $(k,d)$-vertex-arboricity and denoted by $va^{\equiv}_{k,d}(G)$. In this paper, we give sharp upper bounds for $va^{\equiv}_{1,1}(K_{n,n})$ and $va^{\equiv}_{k,\infty}(K_{n,n})$ by showing that $va^{\equiv}_{1,1…
On Almost Well-Covered Graphs of Girth at Least 6
本文研究独立间隙为一的几乎良覆盖图，聚焦围长至少为6的图类，证明其至多有两个与两个叶子相邻的顶点，给出特征刻画并开发多项式时间识别算法。
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…
We provide proofs of the following theorems by considering the entropy of random walks: Theorem 1.(Alon, Hoory and Linial) Let G be an undirected simple graph with n vertices, girth g, minimum degree at least 2 and average degree d: Odd girth: If g=2r+1,then n \geq 1 + d*(\Sum_{i=0}^{r-1}(d-1)^i) Even girth: If g=2r,then n \geq 2*(\Sum_{i=0}^{r-1} (d-1)^i) Theorem 2.(Hoory) Let G = (V_L,V_R,E) be a bipartite graph of girth g = 2r, with n_L = |V_L| and n_R = |V_R|, minimum degree at least 2 and the left and right average degrees d_L and d_R. Then, n_L \geq \Sum_{i=0}^{r-1}(d_R-1)^{i/2}(d_L-1)^{…
Lower bounding the MaxCut of high girth 3-regular graphs using the QAOA
本文通过QAOA算法为高环长3正则图的MaxCut问题提供了新的下界，并展示了其在寻找最大割方面的高效性。
In 1994, Thomassen proved that every planar graph is 5-list-colorable. In 1995, Thomassen proved that every planar graph of girth at least five is 3-list-colorable. His proofs naturally lead to quadratic-time algorithms to find such colorings. Here, we provide the first such linear-time algorithms to find such colorings. For a fixed surface S, Thomassen showed in 1997 that there exists a linear-time algorithm to decide if a graph embedded in S is 5-colorable and similarly in 2003 if a graph of girth at least five embedded in S is 3-colorable. Using the theory of hyperbolic families, the author…
The dominating set problem and its generalization, the distance-$r$ dominating set problem, are among the well-studied problems in the sequential settings. In distributed models of computation, unlike for domination, not much is known about distance-r domination. This is actually the case for other important closely-related covering problem, namely, the distance-$r$ independent set problem. By result of Kuhn et al. we know the distributed domination problem is hard on high girth graphs; we study the problem on a slightly restricted subclass of these graphs: graphs of bounded expansion with hig…
We study homomorphism problems of signed graphs. A signed graph is an undirected graph where each edge is given a sign, positive or negative. An important concept for signed graphs is the operation of switching at a vertex, which is to change the sign of each incident edge. A homomorphism of a graph is a vertex-mapping that preserves the adjacencies; in the case of signed graphs, we also preserve the edge-signs. Special homomorphisms of signed graphs, called s-homomorphisms, have been studied. In an s-homomorphism, we allow, before the mapping, to perform any number of switchings on the source…
Matching Cuts in Graphs of High Girth and H-Free Graphs
该研究解决了匹配割问题的20年公开难题，证明多个匹配割相关问题在高围长有界度二部图、特定无H图上是NP完全的，还得到完美匹配割在无H子图图上的完整复杂度分类。
In various research papers, such as [2], one will find the claim that the minLA is optimally solvable on outerplanar graphs, with a reference to [1]. However, the problem solved in that publication, which we refer to as the planar minimum linear arrangement problem (planar minLA), is different from the minimum linear arrangement problem (minLA), as we show in this article.
On arc-density of pushably $3$-critical oriented graphs
本文证明了除四个例外，n个顶点的可推3-临界有向图至少有(15n+2)/13条弧且界限紧，并据此改进了特定有向图类的可推色数、同态及L(p,q)跨度等已知结果。
Given a graph $G$ of degree $k$ over $n$ vertices, we consider the problem of computing a near maximum cut or a near minimum bisection in polynomial time. For graphs of girth $2L$, we develop a local message passing algorithm whose complexity is $O(nkL)$, and that achieves near optimal cut values among all $L$-local algorithms. Focusing on max-cut, the algorithm constructs a cut of value $nk/4+ n\mathsf{P}_\star\sqrt{k/4}+\mathsf{err}(n,k,L)$, where $\mathsf{P}_\star\approx 0.763166$ is the value of the Parisi formula from spin glass theory, and $\mathsf{err}(n,k,L)=o_n(n)+no_k(\sqrt{k})+n \sq…
A $k$-frugal colouring of a graph $G$ is a proper colouring of the vertices of $G$ such that no colour appears more than $k$ times in the neighbourhood of a vertex. This type of colouring was introduced by Hind, Molloy and Reed in 1997. In this paper, we study the frugal chromatic number of planar graphs, planar graphs with large girth, and outerplanar graphs, and relate this parameter with several well-studied colourings, such as colouring of the square, cyclic colouring, and $L(p,q)$-labelling. We also study frugal edge-colourings of multigraphs.
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