Jieru Feng, Yuping Gao, Jianliang Wu
In 1965, Vizing proved that every planar graph $G$ with maximum degree $Δ\geq 8$ is edge $Δ$-colorable. It is also proved that every planar graph $G$ with maximum degree $Δ=7$ is edge $Δ$-colorable by Sanders and Zhao, independently by Zhang. In this paper, we extend the above results by showing that every $K_5$-minor free graph with maximum degree $Δ$ at least seven is edge $Δ$-colorable.
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We demonstrate that for every positive integer $Δ$, every K\_4-minor-free graph with maximum degree $Δ$ admits an equitable coloring with k colors wherek $\ge$ ($Δ$+3)/2. This bound is tight and confirms a conjecture by Zhang and Whu. We do not use the discharging method but rather exploit decomposition trees of K 4-minor-free graphs.
Strong odd coloring in minor-closed classes
本文研究小闭图类中的强奇着色问题，证明其强奇色数存在常数上界，并对外平面图几乎确定了最小常数。
A total $k$-coloring of a graph $G$ is a coloring of $V(G)\cup E(G)$ using $k$ colors such that no two adjacent or incident elements receive the same color. The total chromatic number $χ"(G)$ of $G$ is the smallest integer $k$ such that $G$ has a total $k$-coloring. In the paper, it is proved that for any $K_5$-minor-free graph $G$, $χ''(G)\leq Δ(G)+2$ if $Δ(G)\geq 7$. Moreover, $χ"(G)=Δ(G)+1$ if $Δ(G)\geq 10$.
It has been conjectured that if a finite graph has a vertex coloring such that the union of any two color classes induces a connected graph, then for every set $T$ of vertices containing exactly one member from each color class there exists a complete minor such that $T$ contains exactly one member from each branching set. Here we prove the statement for line graphs.
A note on extendable sets of colorings and rooted minors
本文将 DeVos 和 Seymour 关于平面图中 3-着色可扩展集合的结果推广到无 $K_{k+2}$ 子式且以 $X$ 为根的无 $K_{k+1}$ 子式图上的 $k$-着色情形。
We prove that every simple connected graph with no $K_5$ minor admits a proper 4-coloring such that the neighborhood of each vertex $v$ having more than one neighbor is not monochromatic, unless the graph is isomorphic to the cycle of length 5. This generalizes the result by S.-J. Kim, S. J. Lee, and W.-J. Park on planar graphs.
Edge-coloring 4- and 5-regular projective planar graphs with no Petersen-minor
本文证明了无彼得森子图的4-和5-正则投影平面图可以被r-边着色。
A clique-coloring of a graph $G$ is a coloring of the vertices of $G$ so that no maximal clique of size at least two is monochromatic. The clique-hypergraph, $\mathcal{H}(G)$, of a graph $G$ has $V(G)$ as its set of vertices and the maximal cliques of $G$ as its hyperedges. A (vertex) coloring of $\mathcal{H}(G)$ is a clique-coloring of $G$. The clique-chromatic number of $G$ is the least number of colors for which $G$ admits a clique-coloring. Every planar graph has been proved to be 3-clique-colorable (Electr. J. Combin. 6 (1999), \#R26). Recently, we showed that every claw-free planar graph…
A fan $F_k$ is a graph that consists of an induced path on $k$ vertices and an additional vertex that is adjacent to all vertices of the path. We prove that for all positive integers $q$ and $k$, every graph with sufficiently large chromatic number contains either a clique of size $q$ or a vertex-minor isomorphic to $F_k$. We also prove that for all positive integers $q$ and $k\ge 3$, every graph with sufficiently large chromatic number contains either a clique of size $q$ or a pivot-minor isomorphic to a cycle of length $k$.
Defective coloring is perfect for minors
该研究证实了子式闭族缺陷色数等于标准构造下界的猜想，得到聚类色数与禁子式树深度的线性关系，改进了此前的指数界。
In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\geq 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. We show that every graph with no $K_t$ minor is $O(t(\log t)^β)$-colorable for every $β> 1/4$, making the first improvement on the order of magnitude of the Kostochka-Thomason bound.
Coloring Small $K_t$-Minor-Free Graphs
本文利用星收缩技术改进了小规模 $K_t$-无 minors 图的着色界，并扩展到所有此类图，将 $O(t\log\log t)$ 改进为 $O(t\sqrt{\log\log t})$。
Strengthening Hadwiger's conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$-minor is properly $(t-1)$-colorable, this is known as the Odd Hadwiger's conjecture. We prove a relaxation of the above conjecture, namely we show that every graph with no odd $K_t$-minor admits a vertex $(2t-2)$-coloring such that all monochromatic components have size at most $\lceil \frac{1}{2}(t-2) \rceil$. The bound on the number of colors is optimal up to a factor of $2$, improves previous bounds for the same problem by Kawarabayashi (2008), Kang and Oum (2019), Liu and Wood (2…
As a strengthening of Hadwiger's conjecture, Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. We prove two weaker variants of this conjecture. Firstly, we show that for each $t \geq 2$, every graph with no odd $K_t$ minor has a partition of its vertex set into $6t-9$ sets $V_1, \dots, V_{6t-9}$ such that each $V_i$ induces a subgraph of bounded maximum degree. Secondly, we prove that for each $t \geq 2$, every graph with no odd $K_t$ minor has a partition of its vertex set into $10t-13$ sets $V_1, \dots, V_{10t-13}$ such that each $V_i$ induces a s…
Clustered Coloring of Graphs Excluding a Subgraph and a Minor
本文研究排除子图和/或子式的图的聚类着色问题，证明了多类图的有界聚类着色数最优界，改进了奇子式相关猜想的现有界值。
Weak diameter choosability of graphs with an excluded minor
本文扩展了Dvořák和Norin的结果，证明了对于任意图H，H-无子式图具有有界弱直径的最优3-可选性，并在H为平面图时证明了更强的预着色扩展结果及指数级着色存在性，同时指出该列表着色结果无法扩展至奇无子式图。
Proper conflict-free list-coloring, odd minors, subdivisions, and layered treewidth
本文研究图的正常无冲突列表着色问题，针对稀疏图与稠密图分别证明了两个逆命题，给出相关问题的部分答案，还得到子式闭族的更优界及分层树宽有界图的着色数上界，可应用于(g,k)-平面图。
A clique-coloring of a given graph $G$ is a coloring of the vertices of $G$ such that no maximal clique of size at least two is monocolored. The clique-chromatic number of $G$ is the least number of colors for which $G$ admits a clique-coloring. It has been proved that every planar graph is $3$-clique colorable and every claw-free planar graph, different from an odd cycle, is $2$-clique colorable. In this paper, we generalize these results to $K_{3,3}$-minor free ($K_{3,3}$-subdivision free) graphs.
Disjoint Correspondence Colorings for $K_5$-Minor-free Graphs
研究$K_5$-无焦图的不相交对应着色，证明了构造3个不相交着色的可能性，并提供了不满足条件的例子。
Centered colorings in minor-closed graph classes
针对minor-封闭图类中的中心着色问题，证明了对固定正整数t，所有不含K_t-子式的图可通过p-中心着色使用O(p^{t-1})种颜色，为该类图的着色提供了复杂度结果。
Hadwiger's conjecture from 1943 states that for every integer $t\ge1$, every graph either can be $t$-colored or has a subgraph that can be contracted to the complete graph on $t+1$ vertices. As pointed out by Paul Seymour in his recent survey on Hadwiger's conjecture, proving that graphs with no $K_7$ minor are $6$-colorable is the first case of Hadwiger's conjecture that is still open. It is not known yet whether graphs with no $K_7$ minor are $7$-colorable. Using a Kempe-chain argument along with the fact that an induced path on three vertices is dominating in a graph with independence numbe…
Proper conflict-free 7-coloring of planar graphs
该研究针对不含K₅、Q₆子式的图，证明其恰当无冲突着色的7色上界，改进平面图此前的8色上界，还推导了不含Kₖ₊₁子式图的奇偶无冲突着色的2k-1色相关结论。
Centered colorings and weak coloring numbers in minor-closed graph classes
本文研究了在排除特定小图的图类中，q-中心色数和q阶弱色数的上界，证明了在O(q)因子内可确定，并在排除平面图时达到常数因子。结果改进了K_t-无小图图的q-中心色数界。
Hadwiger and Hajós conjectured that for every positive integer $t$, $K_{t+1}$-minor free graphs and $K_{t+1}$-topological minor free graphs are properly $t$-colorable, respectively. Clustered coloring version of these two conjectures which only require monochromatic components to have bounded size has been extensively studied. In this paper we consider the clustered coloring version of the immersion-variant of Hadwiger's and Hajós' conjecture proposed by Lescure and Meyniel and independently by Abu-Khzam and Langston. We determine the minimum number of required colors for $H$-immersion free gr…
A signed graph $(G, Σ)$ is a graph $G$ and a subset $Σ$ of its edges which corresponds to an assignment of signs to the edges: edges in $Σ$ are negative while edges not in $Σ$ are positive. A closed walk of a signed graph is balanced if the product of the signs of its edges (repetitions included) is positive, and unbalanced otherwise. The unbalanced-girth of a signed graph is the length of a shortest unbalanced closed walk (if such a walk exists). A homomorphism of $(G,Σ)$ to $(H,Π)$ is a homomorphism of $G$ to $H$ which preserves the balance of closed walks. In this work, given a signed bipar…
On the simultaneous edge coloring of graphs
Given a partial edge coloring of a complete graph $K_n$ and lists of allowed colors for the non-colored edges of $K_n$, can we extend the partial edge coloring to a proper edge coloring of $K_n$ using only colors from the lists? We prove that this question has a positive answer in the case when both the partial edge coloring and the color lists satisfy certain sparsity conditions.
Majority distinguishing edge coloring
本文结合多数着色与区分着色概念，证明无悬挂边图可用至多 ⌈√Δ⌉+5 种颜色实现多数区分边着色，并推广到若干图类及对称有向图。
For $p\in \mathbb{N}$, a coloring $λ$ of the vertices of a graph $G$ is {\em{$p$-centered}} if for every connected subgraph~$H$ of $G$, either $H$ receives more than $p$ colors under $λ$ or there is a color that appears exactly once in $H$. In this paper, we prove that every $K_t$-minor-free graph admits a $p$-centered coloring with $\mathcal{O}(p^{g(t)})$ colors for some function $g$. In the special case that the graph is embeddable in a fixed surface $Σ$ we show that it admits a $p$-centered coloring with $\mathcal{O}(p^{19})$ colors, with the degree of the polynomial independent of the genu…
An $\ell$-facial edge-coloring of a plane graph is a coloring of its edges such that any two edges at distance at most $\ell$ on a boundary walk of any face receive distinct colors. It is the edge-coloring variant of the $\ell$-facial vertex coloring, which arose as a generalization of the well-known cyclic coloring. It is conjectured that at most $3\ell + 1$ colors suffice for an $\ell$-facial edge-coloring of any plane graph. The conjecture has only been confirmed for $\ell \le 2$, and in this paper, we prove its validity for $\ell = 3$.
An l-facial edge coloring of a plane graph is a coloring of the edges such that any two edges at distance at most l on a boundary walk of some face receive distinct colors. It is conjectured that 3l + 1 colors suffice for an l-facial edge coloring of any plane graph. We prove that 7 colors suffice for a 2-facial edge coloring of any plane graph and therefore confirm the conjecture for l = 2.
For a graph $G$ of order $n$ a maximal edge coloring is a proper edge coloring with $χ'(K_n)$ colors such that adding any edge to $G$ in any color makes it improper. Meszka and Tyniec proved that for some values of the number of edges there are no graphs with a maximal edge coloring, while for some other values, they provided constructions of such graphs. However, for many values of the number of edges determining whether there exists any graph with a maximal edge coloring remained open. We give a complete solution of this problem.
We propose the notion of a majority $k$-edge-coloring of a graph $G$, which is an edge-coloring of $G$ with $k$ colors such that, for every vertex $u$ of $G$, at most half the edges of $G$ incident with $u$ have the same color. We show the best possible results that every graph of minimum degree at least $2$ has a majority $4$-edge-coloring, and that every graph of minimum degree at least $4$ has a majority $3$-edge-coloring. Furthermore, we discuss a natural variation of majority edge-colorings and some related open problems.
A $k$-{\it edge-weighting} $w$ of a graph $G$ is an assignment of an integer weight, $w(e)\in \{1,\dots, k\}$, to each edge $e$. An edge weighting naturally induces a vertex coloring $c$ by defining $c(u)=\sum_{u\sim e} w(e)$ for every $u \in V(G)$. A $k$-edge-weighting of a graph $G$ is \emph{vertex-coloring} if the induced coloring $c$ is proper, i.e., $c(u) \neq c(v)$ for any edge $uv \in E(G)$. Given a graph $G$ and a vertex coloring $c_0$, does there exist an edge-weighting such that the induced vertex coloring is $c_0$? We investigate this problem by considering edge-weightings defined o…
We present a novel algorithm for edge-coloring of multigraphs. The correctness of this algorithm for multigraphs with $χ' > Δ+1$ ($χ'$ is the chromatic edge number and $Δ$ is the maximum vertex degree) would prove a long standing conjecture in edge-coloring of multigraphs.
A facial parity edge coloring of a 2-edge connected plane graph is an edge coloring where no two consecutive edges of a facial walk of any face receive the same color. Additionally, for every face f and every color c either no edge or an odd number of edges incident to f are colored by c. Czap, Jendrol', Kardoš and Sotak showed that every 2-edge connected plane graph admits a facial parity edge coloring with at most 20 colors. We improve this bound to 16 colors.
Motivated by the Erdős-Faber-Lovász (EFL) conjecture for hypergraphs, we consider the list edge coloring of linear hypergraphs. We discuss several conjectures for list edge coloring linear hypergraphs that generalize both EFL and Vizing's theorem for graphs. For example, we conjecture that in a linear hypergraph of rank 3, the list edge chromatic number is at most 2 times the maximum degree plus 1. We show that for sufficiently large fixed rank and sufficiently large degree, the conjectures are true.
Normal 6-edge-colorings of cubic graphs with oddness 2
本文推广了Mazzuoccolo和Mkrtchyan关于循环置换图正常6-边着色的结果，证明了每个奇度为2的三次图都允许用至多6种颜色进行正常边着色。
List-Edge-Coloring Triangulations with Maximum Degree at most 5
该研究证明了最大度不超过5的三角剖分满足列表边着色猜想，推进了图论中列表边着色相关问题的研究进展。
We consider the problem of extending and avoiding partial edge colorings of hypercubes; that is, given a partial edge coloring $φ$ of the $d$-dimensional hypercube $Q_d$, we are interested in whether there is a proper $d$-edge coloring of $Q_d$ that agrees with the coloring $φ$ on every edge that is colored under $φ$; or, similarly, if there is a proper $d$-edge coloring that disagrees with $φ$ on every edge that is colored under $φ$. In particular, we prove that for any $d\geq 1$, if $φ$ is a partial $d$-edge coloring of $Q_d$, then $φ$ is avoidable if every color appears on at most $d/8$ edg…
This appendix for our article, "Almost-rainbow edge-colorings of some small subgraphs", contains the full proof of Theorem 4.1.
We consider the following type of question: Given a partial proper $d$-edge coloring of the $d$-dimensional hypercube $Q_d$, and lists of allowed colors for the non-colored edges of $Q_d$,can we extend the partial coloring to a proper $d$-edge coloring using only colors from the lists? We prove that this question has a positive answer in the case when both the partial coloring and the color lists satisfy certain sparsity conditions.
\textit{A star edge coloring} of a graph is a proper edge coloring without bichromatic paths and cycles of length four. In this paper we establish tight upper bounds for trees and subcubic outerplanar graphs, and derive an upper bound for outerplanar graphs.
In the first part, we introduce a notion a degree of edge-colorings of bicubic plane graphs and proves some local formula of the graded number of colorings. In the second part, we give a new proof of a result of Fisk saying that any two edge-3-colorings of a planar bicubic graph are Kempe equivalent. Additionaly we show that the degree of colorings behaves well with respect to Kempe equivalence.
In a proper edge-coloring of a cubic graph, an edge $e$ is normal if the set of colors used by the edges adjacent to $e$ has cardinality 3 or 5. The Petersen coloring conjecture asserts that every bridgeless cubic graph has a normal 5-edge-coloring, that is, a proper 5-edge-coloring such that all edges are normal. In this paper, we prove a result related to the Petersen coloring conjecture. The parameter $μ_3$ is a measurement for cubic graphs, introduced by Steffen in 2015. Our result shows that every bridgeless cubic graph $G$ has a proper 5-edge-coloring such that at least $|E(G)|-μ_3(G)$, …
{\emph A star edge-coloring} of a graph is a proper edge-coloring without bichromatic paths and cycles of length four. In this paper, we consider the list version of this coloring and prove that the list star chromatic index of every subcubic graph is at most $7$, answering the question of Dvořák et al. (Star chromatic index, J. Graph Theory 72 (2013), 313--326).
A balanced edge-coloring of the complete graph is an edge-coloring such that every vertex is incident to each color the same number of times. In this short note, we present a construction of a balanced edge-coloring with six colors of the complete graph on $n=13^k$ vertices, for every positive integer $k$, with no rainbow $K_4$. This solves a problem by Erdős and Tuza.
We propose an effective algorithm that enumerates (and actually finds) all 3-edge colorings and Hamiltonian cycles in a cubic graph. The idea is to make a preliminary run that separates the vertices into two types: ``rigid'' (such that the edges incident to them admit a unique coloring) and ``soft'' ones (such that the edges incident to them admit two distinct colorings), and then to perform the coloring. The computational complexity of this algorithm is on a par with (or even below) the fastest known algorithms that find a single 3-edge coloring or a Hamiltonian cycle for a cubic graph.
Given a 3-colorable graph $X$, the 3-coloring complex $B(X)$ is the graph whose vertices are all the independent sets which occur as color classes in some 3-coloring of $X$. Two color classes $C,D \in V(B(X))$ are joined by an edge if $C$ and $D$ appear together in a 3-coloring of $X$. The graph $B(X)$ is 3-colorable. Graphs for which $B(B(X))$ is isomorphic to $X$ are termed reflexive graphs. In this paper, we consider 3-edge-colorings of cubic graphs for which we allow half-edges. Then we consider the 3-coloring complexes of their line graphs. The main result of the paper is a surprising out…
A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most two receive distinct colors. It is known that every planar graph with maximum degree D has a strong edge coloring with at most 4D + 4 colors. We show that 3D + 6 colors suffice if the graph has girth 6, and 3D colors suffice if the girth is at least 7. Moreover, we show that cubic planar graphs with girth at least 6 can be strongly edge colored with at most 9 colors.
A graph H is t-apex if H-X is planar for some subset X of V(H) of size t. For any integer t>=0 and a fixed t-apex graph H, we give a polynomial-time algorithm to decide whether a (t+3)-connected H-minor-free graph is colorable from a given assignment of lists of size t+4. The connectivity requirement is the best possible in the sense that for every t>=1, there exists a t-apex graph H such that testing (t+4)-colorability of (t+2)-connected H-minor-free graphs is NP-complete. Similarly, the size of the lists cannot be decreased (unless P=NP), since for every t>=1, testing (t+3)-list-colorability…
Motivated by the Erdos-Faber Lovasz conjecture (EFL) for hypergraphs, we explore relationships between several conjectures on the edge coloring of linear hypergraphs. In particular, we are able to increase the class of hypergraphs for which EFL is true.
Clustered Graph Coloring and Layered Treewidth
本文研究图着色中单色分量有界聚类的问题，证明了对于固定整数s,t,k，分层树宽至多k且不含K_{s,t}子图的图是(s+2)-可着色的且聚类有界，推广了平面图等图类的结果。
This paper has been withdrawn by the author due to an error in the proof.
We define a method for edge coloring signed graphs and what it means for such a coloring to be proper. Our method has many desirable properties: it specializes to the usual notion of edge coloring when the signed graph is all-negative, it has a natural definition in terms of vertex coloring of a line graph, and the minimum number of colors required for a proper coloring of a signed simple graph is bounded above by Δ + 1 in parallel with Vizing's Theorem. In fact, Vizing's Theorem is a special case of the more difficult theorem concerning signed graphs.
A cycle is $2$-colored if its edges are properly colored by two distinct colors. A $(d,s)$-edge colorable graph $G$ is a $d$-regular graph that admits a proper $d$-edge coloring in which every edge of $G$ is in at least $s-1$ $2$-colored $4$-cycles. Given a $(d,s)$-edge colorable graph $G$ and a list assigment $L$ of forbidden colors for the edges of $G$ satisfying certain sparsity conditions, we prove that there is a proper $d$-edge coloring of $G$ that avoids $L$, that is, a proper edge coloring $φ$ of $G$ such that $φ(e) \notin L(e)$ for every edge $e$ of $G$.
Vertex-coloring graphs with 4-edge-weightings
针对无孤立边图的顶点着色边权重猜想，研究者证明权重集为{1,2,3,4}时该结论成立，推进了图着色领域相关猜想的研究。
Following Penrose, we introduce a family of graph functions defined in terms of contractions of certain products of symmetric tensors along the edges of a graph. Special cases of these functions enumerate edge colorings and cycles of arbitrary length in graphs (in particular, Hamiltonian cycles).
Semistrong edge coloring and $(0,1)$-relaxed strong edge coloring of graphs
本文通过构造一种同时为半强和(0,1)-松弛强的边着色，证明了除K_{Δ,Δ}外所有最大度Δ≥3的连通图G的半强边色数不超过Δ^2-1，并由此推出其(0,1)-松弛强边色数也不超过Δ^2-1。
A proper edge coloring of a graph is strong if it creates no bichromatic path of length three. It is well known that for a strong edge coloring of a $k$-regular graph at least $2k-1$ colors are needed. We show that a $k$-regular graph admits a strong edge coloring with $2k-1$ colors if and only if it covers the Kneser graph $K(2k-1,k-1)$. In particular, a cubic graph is strongly $5$-edge-colorable whenever it covers the Petersen graph. One of the implications of this result is that a conjecture about strong edge colorings of subcubic graphs proposed by Faudree et al. [Ars Combin. 29 B (1990), …
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