Sebastien Bonduelle, František Kardoš
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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。
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。
An injective coloring of a graph $G$ is an assignment of colors to the vertices of $G$ so that any two vertices with a common neighbor have distinct colors. A graph $G$ is injectively $k$-choosable if for any list assignment $L$, where $|L(v)| \geq k$ for all $v \in V(G)$, $G$ has an injective $L$-coloring. Injective colorings have applications in the theory of error-correcting codes and are closely related to other notions of colorability. In this paper, we show that subcubic planar graphs with girth at least 6 are injectively 5-choosable. This strengthens a result of Lužar, Škrekovski, and T…
In this paper, we address the maximum number of vertices of induced forests in subcubic graphs with girth at least four or five. We provide a unified approach to prove that every 2-connected subcubic graph on $n$ vertices and $m$ edges with girth at least four or five, respectively, has an induced forest on at least $n-\frac{2}{9}m$ or $n-\frac{1}{5}m$ vertices, respectively, except for finitely many exceptional graphs. Our results improve a result of Liu and Zhao and are tight in the sense that the bounds are attained by infinitely many 2-connected graphs. Equivalently, we prove that such gra…
The square of every subcubic planar graph of girth at least 6 is 7-choosable
本文证明围长至少为6的次三次平面图的平方是7-可选择的，即其列表色数不超过7，推广了已知结果。
Impact of local girth on the S-packing coloring of k-saturated subcubic graphs
研究局部圈长对k饱和子三次图S-打包着色能力的影响，通过参数分析和构造验证，揭示饱和度与局部圈长共同决定的可接受S-打包序列。
Domination of subcubic planar graphs with large girth
本文研究大围长次三次平面图的支配数，按顶点度数加权，分别证明围长至少 8 和 9 时的两个新严格上界。
On $(1,2^4)$ and $(1,2^5)$-packing edge-coloring of sparse subcubic graphs
本文针对亚立方平面图，确定了使这类图存在(1,2⁵)-、(1,2⁴)-填充边着色的最小girth阈值的范围，证明两阈值均为有限值。
We study the exact square chromatic number of subcubic planar graphs. An exact square coloring of a graph G is a vertex-coloring in which any two vertices at distance exactly 2 receive distinct colors. The smallest number of colors used in such a coloring of G is its exact square chromatic number, denoted $χ^{\sharp 2}(G)$. This notion is related to other types of distance-based colorings, as well as to injective coloring. Indeed, for triangle-free graphs, exact square coloring and injective coloring coincide. We prove tight bounds on special subclasses of planar graphs: subcubic bipartite pla…
A matching $M$ in a graph $G$ is uniquely restricted if no other matching in $G$ covers the same set of vertices. We prove that any connected subcubic graph with $n$ vertices and girth at least $5$ contains a uniquely restricted matching of size at least $(n-1) / 3$ except for two exceptional cubic graphs of order $14$ and $20$.
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 confirm Jones' Conjecture for subcubic graphs. Namely, if a subcubic planar graph does not contain $k+1$ vertex-disjoint cycles, then it suffices to delete $2k$ vertices to obtain a forest.
On $(1^2,2^2)$-packing edge-coloring of sparse subcubic graphs
本文研究了稀疏次立方图的(1²,2²)-打包边着色问题，证明了所有3-不规则次立方多图均可被此类着色，且该结果最优。同时探讨了平面次立方图的着色性质，证明了足够大的环长可保证(1²,2²)-打包边着色。
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…
Wegner conjectured in 1977 that the square of every planar graph with maximum degree at most $3$ is $7$-colorable. We prove this conjecture using the discharging method and computational techniques to verify reducible configurations.
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…
The square of a subcubic planar graph without a 5-cycle is 7-choosable
本文证明亚立方平面图中不包含5-环的平方是7-可选择的，改进了之前的相关结论。
We strengthen a result of Dross, Montassier and Pinlou (2017) that the vertex set of every triangle-free planar graph can be decomposed into a set that induces a forest and a set that induces a forest with maximum degree at most $5$, showing that $5$ can be replaced by $3$.
$k$-intersection edge-coloring subcubic planar multigraphs
给定一个简单图的边染色，给每个顶点$v$分配一个集合$S_v$，该集合由与$v$关联的边上使用的颜色组成。图的$k$交色指数是使边集能被适当$t$着色的最小$t$，另外还要求对每两个邻顶点$u$和$v$，有$|S_u \cap S_v| \le k$ .对于所有的$k \neq 2$，该值对于次三次可平面图是已知的，而且，这些值是最佳可能的。我们很自然地将这个定义推广到边重数有界的多重图，并且证明了边重数至多为2的次立方平面多重图的2-交色数至多为5，这是一个尖锐的结果。
Large induced forests in planar graphs with girth 4 or 5
本文给出了围长为$4$和$5$的平面图的最大诱导森林的阶的一些新的下界.特别地，我们证明了$n$阶的无三角形平面图存在至少$\frac{6n+7}{11}$阶的诱导森林，改进了Salavatipour [M. R. Salavatipour，Large induced forests in triangle-free planar graphs，Graphs and Combinatorics，22：113-126，2006]。我们还证明了一个阶为$n$且围长至少为$5$的平面图存在一个阶至少为$\frac{44n+50}{69}$的诱导森林。
Splitting Planar Graphs of Girth 6 into Two Linear Forests with Short Paths
最近，Borodin，Kostochka和Yancey（On $1$ -inappropriate $2$ -coloring of sparse graphs. Discrete Mathematics，313（22），2013）示出了围长至少为$7$的每个平面图的顶点可以是$2$着色的，使得每个颜色类导出匹配的子图。我们证明了任何周长至少为$6$的平面图都允许$2$颜色的顶点着色，使得每个单色分量是长度至多为$14$的路。此外，我们显示了这个结果的列表版本。另一方面，对于每个正整数$t\geq 3$，我们构造了一个周长$4$的平面图，使得在$2$颜色的顶点的任何着色中，存在长度至少为$t$的单色路径.围长$5$的每个平面图是否允许没有长单色路的$2$ -染色仍然是开放的。
Partition subcubic planar graphs into independent sets
本文研究次三次平面图的packing $(1^{\ell}, 2^k)$-染色，证明了每个次三次平面图都是packing $(1,2^5)$-可染色的，且该结果是最优且紧的。
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 $198$.
A lower bound on the order of the largest induced forest in planar graphs with high girth
本文给出了高围长平面图中最小反馈点集大小的新的上界。特别地，我们证明了围长为$g$，大小为$m$的平面图的反馈顶点集的大小至多为$\frac{4m}{3g}$，从而改进了$\frac{2m}{g}$的平凡界.证明了具有最大度$3$和最大阶$n$的$2$ -连通图的反馈顶点集的大小至多为$\frac{n+2}{3}$ .
A matching $M$ in a graph $G$ is uniquely restricted if no other matching in $G$ covers the same set of vertices. We conjecture that every connected subcubic graph with $m$ edges and $b$ bridges that is distinct from $K_{3,3}$ has a uniquely restricted matching of size at least $\frac{m+b}{6}$, and we establish this bound with $b$ replaced by the number of bridges that lie on a path between two vertices of degree at most $2$. Moreover, we prove that every connected subcubic graph of order $n$ and girth at least $7$ has a uniquely restricted matching of size at least $\frac{n-1}{3}$, which part…
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.
The square of every subcubic planar graph without 4-cycles and 5-cycles is 7-choosable
本文证明了不含4-圈和5-圈的次三次平面图的平方是7-可选的，改进了Kim和Lian关于围长至少为6的次三次平面图的结果。
Strong chromatic index of subcubic planar multigraphs
一个重图的强色指数是使其边集可以是$k$ -着色的最小$k$，要求每个色类诱导一个匹配.证明了Faudree，Gyárfás，Schelp和Tuza的一个猜想，证明了最大度不超过3的平面重图的强色指数不超过9，这是一个尖锐的猜想.
A graph $H$ is an immersion of a graph $G$ if $H$ can be obtained by some sugraph $G$ after lifting incident edges. We prove that there is a polynomial function $f:\Bbb{N}\times\Bbb{N}\rightarrow\Bbb{N}$, such that if $H$ is a connected planar subcubic graph on $h>0$ edges, $G$ is a graph, and $k$ is a non-negative integer, then either $G$ contains $k$ vertex/edge-disjoint subgraphs, each containing $H$ as an immersion, or $G$ contains a set $F$ of $f(k,h)$ vertices/edges such that $G\setminus F$ does not contain $H$ as an immersion.
Induced Matchings in Subcubic Graphs
证明了一个具有$m$边的三次图至少有$m/9$边的诱导匹配。我们的结果推广了Kang，Mnich和Müller关于可平面图的结果（Induced matching in subcubic可平面图，SIAM J. Discrete Math.26（2012）1383-1411），并解决了Henning和Rautenbach的猜想（Induced matching in subcubic graphs without short cycles，to appear in Discrete Math.）。
The Coloring Game on Planar Graphs with Large Girth, by a result on Sparse Cactuses
我们用$χ$ g（G）表示图G的游戏色数，它是Alice在G上赢得着色游戏所需的最小颜色数.我们从Montassier et al. [M. Montassier，P. Ossona de Mendez，A. Raspaud和X. Zhu，Decomposing a graph into forests，J. Graph Theory Ser. B，102（1）：38-52，2012]，并且独立地，从Wang和Zhang，[Y. Wang和Q.张某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.人们可能会问，对于足够大的围长，这个5的界是否可以改进。在本文中，我们证明它不能。不仅如此，我们还证明了存在仙人掌CT（即每条边最多只属于一个圈的图）的$χ$ g（CT）= 5，尽管其圈长任意大，甚至圈间距离任意大.
Locally irregular edge-coloring of subcubic graphs
本文研究次立方图的局部不规则边染色问题，证明最大度为3的可分解无爪图、循环置换图和广义Petersen图至多需要3种颜色，并给出需要3种颜色的围长4三次图无限族。
A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $χ_p(G)$, of a graph $G$ is the minimum $k$ such that $G$ has a packing $k$-coloring. Sloper showed that there are $4$-regular graphs with arbitrarily large packing chromatic number. The question whether the packing chromatic number of subcubic graphs is bounded appears in several papers. We answer this question in the negative. Moreover, we show that for every fixed $k$ a…
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 $Δ\geq 4$.
List-coloring the Square of a Subcubic Graph
图$G$的{ \em平方} $G^2$是具有与$G$相同的顶点集并且两个顶点相邻的图，如果它们在$G$中的距离至多为2。证明了具有最大度$Δ(G)=3$的平面图$G$满足$χ(G^2)\leq 7$ . Kostochka和Woodall证明了对任何图，$G^2$的列表色数等于$G^2$的色数，即对所有的$G$都是$χ_l(G^2)=χ(G^2)$.如果这个猜想成立，则这个猜想（连同Kassen的结果）意味着每个具有$Δ(G)=3$的平面图$G$满足$χ_l(G^2)\leq 7$。我们证明了每个连通图（不一定是平面的）与$Δ(G)=3$以外的彼得森图满足$χ_l(G^2)\leq 8$（这是最好的可能）。另外，我们证明了如果$G$是一个具有$Δ(G)=3$和围长$g(G)\geq 7$的平面图，那么$χ_l(G^2)\leq 7$ . Dvoák、Škrekovski和Tancer证明了，如果$G$是一个平面图，其周长为$Δ(G) = 3$和$g(G) \geq 10$，则有$χ_l(G^2)\leq 6$。我们改进了围长界，证明了如果$G$是一个平面图，且$Δ(G)=3$和$g(G) \geq 9$都有，则$χ_l(G^2) \leq 6$ .我们所有的证明可以很容易地转化为线性时间着色算法。
The Alon-Tarsi Number of Squares of Subcubic Planar Graphs without Cycles of Lengths $4$ to $8$
研究平面图平方的阿隆 - 塔尔西数，通过图多项式定义，Kim和Luo证明特定次立方平面图平方的列表色数≤6，本文强化结果，证明此类图平方的阿隆 - 塔尔西数≤6。
Squares of subcubic planar graphs without cycles of length 4-8 are 6-choosable
本文证明亚立方平面图中不含长度4-8的环时，其平方图的列表色数不超过6。
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…
In 1991, Bollobás and Frieze showed that the threshold for $G_{n,p}$ to contain a spanning maximal planar subgraph is very close to $p = n^{-1/3}$. In this paper, we compute similar threshold ranges for $G_{n,p}$ to contain a maximal bipartite planar subgraph and for $G_{n,p}$ to contain a maximal planar subgraph of fixed girth $g$.
Walk-powers and homomorphism bound of planar graphs
作为四色定理的一个推广，证明了每一个奇围长至少为$2k+1$的可平面图都与$PC_{2k}=(\mathbb{Z}_2^{2k}, \{e_1, e_2, ...,e_{2k}, J\})$有一个同态，其中$e_i$是标准基，$J$是全1向量.注意到，$PC_{2k}$本身是奇数围长$2k+1$，在这项工作中，我们表明，如果猜想是真的，那么$PC_{2k}$是一个最佳的这样的图无论是关于顶点数和边数。利用图的行走幂及其团数的概念得到了这个结果。对于不平衡围长$2k$的二部带号可平面图也证明了类似的结果。本文的工作是在平面一致符号图的统一框架上进行的。
For a positive integer $k$, the $k$-recolouring graph of a graph $G$ has as vertex set all proper $k$-colourings of $G$ with two $k$-colourings being adjacent if they differ by the colour of exactly one vertex. A result of Dyer et al. regarding graphs of bounded degeneracy implies that the $7$-recolouring graphs of planar graphs, the $5$-recolouring graphs of triangle-free planar graphs and the $4$-recolouring graphs planar graphs of girth at least six are connected. On the other hand, there are planar graphs whose $6$-recolouring graph is disconnected, triangle-free planar graphs whose $4$-re…
Hypohamiltonian planar cubic graphs with girth five
如果一个图不是哈密尔顿图，但如果去掉任何一个顶点，它就变成哈密尔顿图，则称它是次哈密尔顿图。从1981年Jassen构造平面三次图开始，人们发现了许多次哈密尔顿平面三次图。然而，到目前为止发现的所有例子都有4个周期。在这篇笔记中，我们提出了第一个例子，亚哈密尔顿平面三次图的循环连通度为5，从而围长为5。通过计算机搜索，我们证明了这类图的最小成员是三个有76个顶点的图。
List coloring the square of sparse graphs with large degree
我们考虑有界最大平均度图的平方着色问题，即在保证相邻或有共同邻居的两个顶点接收不同颜色的情况下对顶点着色的问题。Borodin等人在2004年和2008年证明了围长至少为7且最大度足够大的平面图$Δ$的平方是列表$(Δ+1)$ -可染的，而围长为6且最大度任意大的平面图的平方不是列表$(Δ+1)$ -可染的。根据欧拉公式，围长至少为$6$的平面图的最大平均度小于$3$，围长至少为$7$的平面图的最大平均度小于$14/5<3$ .我们加强了他们的结果，证明了存在一个函数$f$使得任何具有最大平均度$m<3$和最大度$Δ\geq f(m)$的图的平方都是列表$(Δ+1)$ -可染的. $3$的这一界是最优的，在这个意义上，上述平面图的周长$6$有最大平均度小于$3$和任意大的最大度，而他们的平方不能是$(Δ+1)$ -着色。这同样适用于列表单射$Δ$ -着色。
A graph is $(d_1, \ldots, d_k)$-colorable if its vertex set can be partitioned into $k$ nonempty subsets so that the subgraph induced by the $i$th part has maximum degree at most $d_i$ for each $i\in\{1, \ldots, k\}$. It is known that for each pair $(d_1, d_2)$, there exists a planar graph with girth $4$ that is not $(d_1, d_2)$-colorable. This sparked the interest in finding the pairs $(d_1, d_2)$ such that planar graphs with girth at least $5$ are $(d_1, d_2)$-colorable. Given $d_1\leq d_2$, it is known that planar graphs with girth at least $5$ are $(d_1, d_2)$-colorable if either $d_1\geq …
11/4-colorability of subcubic triangle-free graphs
证明了除两个例外外，所有连通次立方无三角形图的分数色数不超过11/4，且该界紧，改进了平面图情形。研究问题为分数色数上界，方法为构造性证明，贡献在于紧界及例外刻画。
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 coloring of a graph is a proper edge coloring in which every color class is an induced matching. The strong chromatic index $χ_s'(G)$ of a graph $G$ is the minimum number of colors in a strong edge coloring of $G$. Let $Δ\geq 4$ be an integer. In this note, we study the odd graphs and show the existence of some special walks. By using these results and Chang's ideas in [Discuss. Math. Graph Theory 34 (4) (2014) 723--733], we show that every planar graph with maximum degree at most $Δ$ and girth at least $10 Δ- 4$ has a strong edge coloring with $2Δ- 1$ colors. In addition, we pro…
Strong edge-colouring of sparse planar graphs
图的强边染色是每一个色类诱导一个匹配的真边染色。我们知道，每个具有最大度$Δ$的平面图都有强边染色，最多有$4Δ+4$色。我们表明，$3Δ+1$颜色足够，如果图的围长为6，和$4Δ$颜色足够，如果$Δ\geq 7$或围长至少是5。在本文的最后一部分，我们提出了一些问题有关的一个长期的猜想Vizing正常边染色的平面图。
We prove a conjecture of Dvořák, Král, Nejedlý, and Škrekovski that planar graphs of girth at least five are square $(Δ+2)$-colorable for large enough $Δ$. In fact, we prove the stronger statement that such graphs are square $(Δ+2)$-choosable and even square $(Δ+2)$-paintable.
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…
In 1994, Thomassen famously proved that every planar graph is 5-choosable, resolving a conjecture initially posed by Vizing and, independently, Erd\H{os}, Rubin, and Taylor in the 1970s. Later, Thomassen proved that every planar graph of girth at least five is 3-choosable. In this paper, we introduce the concept of a \emph{local girth list assignment}: a list assignment wherein the list size of a vertex depends not on the girth of the graph, but rather on the length of the shortest cycle in which the vertex is contained. We give a local list colouring theorem unifying the two theorems of Thoma…
Choosability of planar graphs of girth 5
Rumassen证明了围长为5的平面图是列表可着色的，使得所有顶点都有大小为2或3的列表，并且列表大小为2的顶点都与外表面关联，形成一个独立集。我们提出了一个加强这一结果，放松限制的顶点与列表的大小为2。这个结果被用来限制具有一个预着色面的3-列表着色临界平面图的大小。
On 2-Distance ($Δ+4$)-coloring of planar graphs with girth at least five
该研究针对围长不小于5的平面图，证明了当最大度$Δ\geq 22$时，这类图可使用$Δ+4$种颜色完成2-距离着色，推进了平面图2-距离着色的相关理论研究。
Exponential Domination in Subcubic Graphs
作为图中支配的自然变体，Dankelmann等人[Domination with exponential decay，Discrete Math.309（2009）5877-5883]引入了指数支配，其中顶点被认为具有随距离呈指数下降的某种支配功率，并且被支配的顶点必须积累足够量的从支配顶点发出的这种功率。更精确地说，如果$S$是图$G$的顶点集，则$S$是$G$的指数支配集，如果对于$V(G)\setminus S$中的每个顶点$u$都有$\sum\limits_{v\in S}\left(\frac{1}{2}\right)^{{\rm dist}_{(G,S)}(u,v)-1}\geq 1$，其中${\rm dist}_{(G,S)}(u,v)$是图$G-(S\setminus \{ v\})$中$u\in V(G)\setminus S$和$v\in S$之间的距离。$G$的指数控制数$γ_e(G)$是$G$的指数控制集的最小阶。本文研究了次三次图的指数控制。我们的结果如下：若$G$是$n(G)$阶的连通次三次图，则有$$\frac{n(G)}{6\log_2(n(G)+2)+4}\leq γ_e(G)\leq \frac{1}{3}(n(G)+2).$$对于每一个$ε>0$，存在某个$g$使得对于每一个围长为$G$的三次图，$γ_e(G)\leq εn(G)$至少有$g$…
On the Strong Chromatic Index of Sparse Graphs
图$G$的强色指数，记为$χ_s'(G)$，是对$G$进行边着色所需的最少颜色数，使得距离至多为2的边接收不同的颜色。表示为$χ_{s,\ell}'(G)$的强列表色度索引是最小整数$k$，使得如果将大小为$k$的任意列表分配给每个边缘，则$G$可以从距离至多为2的边缘接收不同颜色的那些列表中进行边缘着色。本文利用放电方法、组合零点定理和计算证明了：如果$G$是一个次三次平面图，且有$\operatorname{girth}(G) \geq 41$，则有$χ_{s,\ell}'(G) \leq 5$，从而回答了Borodin和Ivanova [Precise upper bounding for the strong edge chromatic number of sparse planar graphs，Discuss.图论，33（4），（2014）759- 770]。我们进一步证明了如果$G$是一个次立方平面图，并且$\operatorname{girth}(G) \geq 30$，那么$χ_s'(G) \leq 5$，改进了同一篇文章中的一个界。最后，如果$G$是一个最大度不超过4的平面图，且$\operatorname{girth}(G) \geq 28$，则有$χ_s'(G) \leq 7$，改进了Wang和Zhao在[Odd graphs and its applic…
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。
An improved bound for 2-distance coloring of planar graphs with girth six
针对围长至少为6且最大度$Δ\neq6$的平面图，证明其2-距离色数$χ_2(G)\neqΔ+4$，改进了该类图2-距离着色的已知最优上界。
Partitions of planar (oriented) graphs into a connected acyclic and an independent set
围绕欧拉有向平面图的顶点划分交叉问题，研究CAI-划分的存在性，证明两类图存在该划分，并构造反例将已有结果推广至有向场景。
On the precise value of the strong chromatic-index of a planar graph with a large girth
图$G$的强$k$ -边染色是从$E(G)$到$\{1,2,\ldots,k\}$的映射，使得距离至多为2的每对不同的边接收不同的颜色。图$G$的强色指数$χ'_s(G)$是使$G$具有强$k$ -边染色的最小$k$.表示为$σ(G)=\max_{xy\in E(G)}\{\operatorname{deg}(x)+\operatorname{deg}(y)-1\}$。很容易看出，对于任何图$G$，都有$σ(G) \le χ'_s(G)$，并且当$G$是树时等式成立。对于最大度为$Δ$的可平面图$G$，利用四色定理证明了$χ'_s(G) \le 4 Δ+4$.然后在不同的$Δ$和周长条件下，将上限简化为$4Δ$，$3Δ+5$，$3Δ+1$，$3Δ$，$2Δ-1$。本文证明了：如果可平面图$G$的围长足够大，且是$σ(G)\geq Δ(G)+2$，则$G$的强色指数恰是$σ(G)$ .这个结果反映了一种直觉，即一个局部围长较大的可平面图看起来像一棵树。
Perfect matchings in planar 3-connected graphs: girth five and minimum degree five
该论文回答了Goedgebeur等人关于平面3-连通图完美匹配的两个问题，构造了围长5且完美匹配数为常数的无限族图，还证明了最小度≥5的此类图的完美匹配数下界。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。