Appearing in different format, Gupta\,(1967), Goldberg\,(1973), Andersen\,(1977), and Seymour\,(1979) conjectured that if $G$ is an edge-$k$-critical graph with $k \ge Δ+1$, then $|V(G)|$ is odd and, for every edge $e$, $E(G-e)$ is a union of disjoint near-perfect matchings, where $Δ$ denotes the maximum degree of $G$. Tashkinov tree method shows that critical graphs contain a subgraph with two important properties named closed and elementary. Recently, efforts have been made in extending graphs beyond Tashkinov trees. However, these results can only keep one of the two essential properties. In this paper, we developed techniques to extend Tashkinov trees to larger subgraphs with both properties. Applying our result, we have improved almost all known results towards Goldberg's conjecture. In particular, we showed that Goldberg's conjecture holds for graph $G$ with $|V(G)| \le 39$ and $|Δ(G)| \le 39$ and Jacobsen's equivalent conjecture holds for $m \le 39$ while the previous known bound is $23$.
For a multigraph G, the integer round-up phi(G) of the fractional chromatic index yields a good general lower bound for the chromatic index . For an upper bound, Kahn showed that for any real c > 0 there exists a positive integer N so that the chromatic index is less than (1+c)*phi(G) whenever the fractional index > N. We show the amount by which the chromatic index can surpass phi(G) is in fact logarithmic, by showing that for any multigraph G with order n > 3 and at least one edge, the chromatic index is less than phi(G) + log (min {(n+1)/3, phi(G)}) .
The strong chromatic index of a multigraph is the minimum $k$ such that the edge set can be $k$-colored requiring that each color class induces a matching. We verify a conjecture of Faudree, Gyárfás, Schelp and Tuza, showing that every planar multigraph with maximum degree at most 3 has strong chromatic index at most 9, which is sharp.
The star chromatic index of a multigraph $G$, denoted $χ'_{s}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star $k$-edge-colorable if $χ'_{s}(G)\le k$. Dvořák, Mohar and Šámal [Star chromatic index, J. Graph Theory 72 (2013), 313--326] proved that every subcubic multigraph is star $7$-edge-colorable. They conjectured in the same paper that every subcubic multigraph should be star $6$-edge-colorable. In this paper, we first prove that it is NP-complete to determine whether $χ'_s(G)\le3…
We prove that for a line perfect multigraph the chromatic index is equal to the list chromatic index. This is a generalization of Galvin's result on bipartite multigraphs. Soon after the first version was submitted to arxiv, I found out that the same result has already been achieved in an older paper by Peterson and Woodall, by a similar but not entirely the same method.
The Goldberg-Seymour Conjecture for $f$-colorings states that the $f$-chromatic index of a loopless multigraph is essentially determined by either a maximum degree or a maximum density parameter. We introduce an oriented version of $f$-colorings, where now each color class of the edge-coloring is required to be orientable in such a way that every vertex $v$ has indegree and outdegree at most some specified values $g(v)$ and $h(v)$. We prove that the associated $(g,h)$-oriented chromatic index satisfies a Goldberg-Seymour formula. We then present simple applications of this result to variations…
Given an edge-coloring of a simple graph, assign to every vertex $v$ a set $S_v$ comprised of the colors used on the edges incident to $v$. The $k$-intersection chromatic index of a graph is the minimum $t$ such that the edge set can be properly $t$-colored, additionally requiring that for every two adjacent vertices $u$ and $v$, $|S_u \cap S_v| \le k$. For all $k \neq 2$, this value is known for subcubic planar graphs, and furthermore, these values are best possible. We naturally extend this definition to multigraphs with bounded edge multiplicity, and we show that every subcubic planar multi…
A locally irregular multigraph is a multigraph whose adjacent vertices have distinct degrees. The locally irregular edge coloring is an edge coloring of a multigraph $G$ such that every color induces a locally irregular submultigraph of $G$. We say that a multigraph $G$ is locally irregular colorable if it admits a locally irregular edge coloring and we denote by ${\rm lir}(G)$ the locally irregular chromatic index of $G$, which is the smallest number of colors required in a locally irregular edge coloring of a locally irregular colorable multigraph $G$. We conjecture that for every connected …
The star chromatic index of a multigraph $G$, denoted $χ'_{s}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star $k$-edge-colorable if $χ'_{s}(G)\le k$. Dvořák, Mohar and Šámal [Star chromatic index, J Graph Theory 72 (2013), 313--326] proved that every subcubic multigraph is star $7$-edge-colorable, and conjectured that every subcubic multigraph should be star $6$-edge-colorable. Kerdjoudj, Kostochka and Raspaud considered the list version of this problem for simple graphs and proved …
Let $G=(V(G), E(G))$ be a multigraph with maximum degree $Δ(G)$, chromatic index $χ'(G)$ and total chromatic number $χ''(G)$. The Total Coloring conjecture proposed by Behzad and Vizing, independently, states that $χ''(G)\leq Δ(G)+μ(G) +1$ for a multigraph $G$, where $μ(G)$ is the multiplicity of $G$. Moreover, Goldberg conjectured that $χ''(G)=χ'(G)$ if $χ'(G)\geq Δ(G)+3$ and noticed the conjecture holds when $G$ is an edge-chromatic critical graph. By assuming the Goldberg-Seymour conjecture, we show that $χ''(G)=χ'(G)$ if $χ'(G)\geq \max\{ Δ(G)+2, |V(G)|+1\}$ in this note. Consequently, $χ'…
A Kneser representation KG(H) for a graph G is a bijective assignment of hyperedges of a hypergraph H to the vertices of G such that two vertices of G are adjacent if and only if the corresponding hyperedges are disjoint. In this paper, we introduce a colored version of the Turan number and use that to determine the chromatic number of some families of graphs in terms of the generalized Turan number of graphs. In particular, we determine the chromatic number of every Kneser multigraph KG(H), where the vertex set of H is the edge set of a multigraph G such that the multiplicity of each edge is …
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.
The rich-neighbor chromatic index is linear in the maximum degree
Keio University（庆应义塾大学）；Ss. Cyril and Methodius University（西里尔和梅托迪乌斯大学）；University of Split（斯普利特大学）；University of Novo Mesto（新梅斯托大学）；University of Ljubljana（卢布尔雅那大学）；Rudolfovo – Science and Technology Centre（鲁道福夫科学与技术中心）
We investigate the effect of a fixed forbidden clique minor upon the strong chromatic index, both in multigraphs and in simple graphs. We conjecture for each $k\ge 4$ that any $K_k$-minor-free multigraph of maximum degree $Δ$ has strong chromatic index at most $\frac32(k-2)Δ$. We present a construction certifying that if true the conjecture is asymptotically sharp as $Δ\to\infty$. In support of the conjecture, we show it in the case $k=4$ and prove the statement for strong clique number in place of strong chromatic index. By contrast, we make a basic observation that for $K_k$-minor-free simpl…
While solving a question on list coloring of planar graphs, Dvořák and Postle introduced the new notion of DP-coloring (they called it correspondence coloring). A DP-coloring of a graph $G$ reduces the problem of finding a coloring of $G$ from a given list $L$ to the problem of finding a "large" independent set in an auxiliary graph $H(G,L)$ with vertex set $\{(v,c)\,: \, v\in V(G) \text{ and } {c\in L(v)} \}$. It is similar to the old reduction by Plesnevič and Vizing of the $k$-coloring problem to the problem of finding an independent set of size $|V(G)|$ in the Cartesian product $G\square K…
In this paper, we prove an enhanced version of the Erdős-Lovász Tihany Conjecture for line graphs of multigraphs. That is, for every graph $G$ whose chromatic number $χ(G)$ is more than its clique number $ω(G)$ and for nonnegative integer $\ell$, any two integers $s,t \geq 3.5\ell+2$ with $s+t = χ(G)+1$, there is a partition $(S,T)$ of the vertex set $V(G)$ such that $χ(G[S])\geq s$ and $χ(G[T])\geq t+\ell$. In particular, when $\ell=1$, we can obtain the same result just for any $s,t\geq4$. The Erdős-Lovász Tihany conjecture is a special case when $\ell=0$.
A multigraph is locally irregular if the degrees of the end-vertices of every multiedge are distinct. The locally irregular coloring is an edge coloring of a multigraph $G$ such that every color induces a locally irregular submultigraph of $G$. A locally irregular colorable multigraph $G$ is any multigraph which admits a locally irregular coloring. We denote by ${\rm lir}(G)$ the locally irregular chromatic index of a multigraph $G$, which is the smallest number of colors required in the locally irregular coloring of the locally irregular colorable multigraph $G$. In case of graphs the definit…
Given a multigraph $G=(V,E)$, the {\em edge-coloring problem} (ECP) is to color the edges of $G$ with the minimum number of colors so that no two adjacent edges have the same color. This problem can be naturally formulated as an integer program, and its linear programming relaxation is called the {\em fractional edge-coloring problem} (FECP). In the literature, the optimal value of ECP (resp. FECP) is called the {\em chromatic index} (resp. {\em fractional chromatic index}) of $G$, denoted by $χ'(G)$ (resp. $χ^*(G)$). Let $Δ(G)$ be the maximum degree of $G$ and let \[Γ(G)=\max \Big\{\frac{2|E(…
Let $G$ be a graph with maximum degree $Δ(G)$ and maximum multiplicity $μ(G)$. Vizing and Gupta, independently, proved in the 1960s that the chromatic index of $G$ is at most $Δ(G)+μ(G)$. The distance between two edges $e$ and $f$ in $G$ is the length of a shortest path connecting an endvertex of $e$ and an endvertex of $f$. A distance-$t$ matching is a set of edges having pairwise distance at least $t$. Edwards et al. proposed the following conjecture: For any graph $G$, using the palette $\{1, \dots, Δ(G)+μ(G)\}$, any precoloring on a distance-$2$ matching can be extended to a proper edge co…
The star chromatic index of a multigraph $G$, denoted $χ'_{st}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bicolored. We survey the results of determining the star chromatic index, present the interesting proofs and techniques, and collect many open problems and conjectures.
In this paper, we present some properties on chromatic polynomials of hypergraphs which do not hold for chromatic polynomials of graphs. We first show that chromatic polynomials of hypergraphs have all integers as their zeros and contain dense real zeros in the set of real numbers. We then prove that for any multigraph $G=(V,E)$, the number of totally cyclic orientations of $G$ is equal to the value of $|P(H,-1)|$, where $P(H,λ)$ is the chromatic polynomial of a hypergraph $H$ which is constructed from $G$. Finally we show that the multiplicity of root "$0$" of $P(H,λ)$ may be at least $2$ for…
We investigate which graphs H have the property that in every graph with bounded clique number and sufficiently large chromatic number, some induced subgraph is isomorphic to a subdivision of H. In an earlier paper, one of us proved that every tree has this property; and in another earlier paper with M. Chudnovsky, we proved that every cycle has this property. Here we give a common generalization. Say a banana is the union of a set of paths all with the same ends but otherwise disjoint. We prove that if H is obtained from a tree by replacing each edge by a banana then H has the property mentio…
Let G = (V, E) be a multigraph without loops and for any x {\in}V let E(x) be the set of edges of G incident to x. A homogeneous edge-coloring of G is an assignment of an integer m >= 2 and a coloring c:E {\to} S of the edges of Gsuchthat|S| = mandforanyx{\in}V,if|E(x)| = mqx+rx with0 <= rx <m, there exists a partition of E(x) in rx color classes of cardinality qx + 1 and other m-rx color classes of cardinality qx. The homogeneous chromatic index \c{hi}(G) is the least m for which there exists such a coloring. We determine \c{hi}(G) in the case that G is a complete multigraph, a tree or a comp…
The arboricity of a graph G is the minimum number of colours needed to colour the edges of G so that every cycle gets at least two colours. Given a positive integer p, we define the generalized p-arboricity Arb_p(G) of a graph G as the minimum number of colours needed to colour the edges of a multigraph G in such a way that every cycle C gets at least min(|C|; p + 1) colours. In the particular case where G has girth at least p + 1, Arb_p(G) is the minimum size of a partition of the edge set of G such that the union of any p parts induce a forest. If we require further that the edge colouring b…
In the 70s, Goldberg, and independently Seymour, conjectured that for any multigraph $G$, the chromatic index $χ'(G)$ satisfies $χ'(G)\leq \max \{Δ(G)+1, \lceilρ(G)\rceil\}$, where $ρ(G)=\max \{\frac {e(G[S])}{\lfloor |S|/2\rfloor} \mid S\subseteq V \}$. We show that their conjecture (in a stronger form) is true for random multigraphs. Let $M(n,m)$ be the probability space consisting of all loopless multigraphs with $n$ vertices and $m$ edges, in which $m$ pairs from $[n]$ are chosen independently at random with repetitions. Our result states that, for a given $m:=m(n)$, $M\sim M(n,m)$ typical…
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 order to solve a question on list coloring of planar graphs, Dvořák and Postle introduced the concept of so called DP-coloring, thereby extending the concept of list-coloring. DP-coloring was anaylized in detail by Bernshteyn, Kostochka, and Pron for graphs and multigraphs; they characterized DP-degree colorable multigraphs and deduced a Brooks' type result from this. The characterization of the corresponding 'bad' covers was later given by Kim and Ozeki. In this paper, the concept of DP-colorings is extended to hypergraphs having multiple (hyper-)edges. We characterize the DP-degree colora…
DP-coloring is a generalization of list coloring that was introduced in 2015 by Dvořák and Postle. The chromatic polynomial of a graph $G$, denoted $P(G,m)$, is equal to the number of proper $m$-colorings of $G$. A well-known tool for computing the chromatic polynomial of graph $G$ is the deletion-contraction formula which relates $P(G,m)$ to the chromatic polynomials of two smaller graphs. The DP color function of a graph $G$, denoted $P_{DP}(G,m)$, is a DP-coloring analogue of the chromatic polynomial, and $P_{DP}(G,m)$ is the minimum number of DP-colorings of $G$ over all possible $m$-fold …
Mkrtchyan and Steffen [J. Graph Theory, 70 (4), 473--482, 2012] showed that every class II simple graph can be decomposed into a maximum $Δ$-edge-colorable subgraph and a matching. They further conjectured that every graph $G$ with chromatic index $Δ(G)+k$ ($k\geq 1$) can be decomposed into a maximum $Δ(G)$-edge-colorable subgraph (not necessarily class I) and a $k$-edge-colorable subgraph. In this paper, we first generalize their result to multigraphs and show that every multigraph $G$ with multiplicity $μ$ can be decomposed into a maximum $Δ(G)$-edge-colorable subgraph and a subgraph with ma…
It is known that every graph of sufficiently large chromatic number and bounded clique number contains, as an induced subgraph, a subdivision of any fixed forest, and a subdivision of any fixed cycle. Equivalently, forests and triangles are pervasive, where H is pervasive (in some class of graphs) if for all s>0, every graph in the class with bounded clique number and sufficiently large chromatic number contains an induced subdivision of H, with every edge subdivided at least s times. Which other graphs are pervasive? Chalopin, Esperet, Li and Ossona de Mendez proved that every such graph is a…
The star chromatic index of a multigraph $G$, denoted by $χ_{\mathrm{star}}'(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length $4$ is bicolored. In this paper, we study the star edge-coloring of Halin graphs, $k$-power graphs and the generalized Petersen graphs $P(3n, n)$.
We present a necessary and sufficient condition for a graph of odd-girth $2k+1$ to bound the class of $K_4$-minor-free graphs of odd-girth (at least) $2k+1$, that is, to admit a homomorphism from any such $K_4$-minor-free graph. This yields a polynomial-time algorithm to recognize such bounds. Using this condition, we first prove that every $K_4$-minor free graph of odd-girth $2k+1$ admits a homomorphism to the projective hypercube of dimension $2k$. This supports a conjecture of the third author which generalizes the four-color theorem and relates to several outstanding conjectures such as Se…
An equitable coloring of a graph $G$ is a proper vertex coloring of $G$ such that the sizes of any two color classes differ by at most one. In the paper, we pose a conjecture that offers a gap-one bound for the smallest number of colors needed to equitably color every block graph. In other words, the difference between the upper and the lower bounds of our conjecture is at most one. Thus, in some sense, the situation is similar to that of chromatic index, where we have the classical theorem of Vizing and the Goldberg conjecture for multigraphs. The results obtained in the paper support our con…
Stability with minuscule structure for chromatic thresholds
KAIST（韩国科学技术院）；Institute for Basic Science (IBS)（基础科学研究院）；Shandong University（山东大学）；Universitat Politècnica de Catalunya (UPC)（加泰罗尼亚理工大学）；Ningbo University（宁波大学）
Institute of Mathematics of the Polish Academy of Sciences（波兰科学院数学研究所）；Institute of Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University（雅盖隆大学数学与计算机科学学院理论计算机科学研究所）；AGH University of Krakow, Faculty of Applied Mathematics（克拉科夫AGH科技大学应用数学学院）
We call a multigraph $(k,d)$-edge colourable if its edge set can be partitioned into $k$ subgraphs of maximum degree at most $d$ and denote as $χ'_{d}(G)$ the minimum $k$ such that $G$ is $(k,d)$-edge colourable. We prove that for every integer $d$, every multigraph $G$ with maximum degree $Δ$ is $(\lceil \fracΔ{d} \rceil, d)$-edge colourable if $d$ is even and $(\lceil \frac{3Δ- 1}{3d - 1} \rceil, d)$-edge colourable if $d$ is odd and these bounds are tight. We also prove that for every simple graph $G$, $χ'_{d}(G) \in \{ \lceil \fracΔ{d} \rceil, \lceil \frac{Δ+1}{d} \rceil \}$ and characteri…
We consider precolouring extension problems for proper edge-colourings of graphs and multigraphs, in an attempt to prove stronger versions of Vizing's and Shannon's bounds on the chromatic index of (multi)graphs in terms of their maximum degree $Δ$. We are especially interested in the following question: when is it possible to extend a precoloured matching to a colouring of all edges of a (multi)graph? This question turns out to be related to the notorious List Colouring Conjecture and other classic notions of choosability.
Resolving a conjecture of Bollobás and Erdős, Gyárfás proved that every graph $G$ of chromatic number $k+1\geq 3$ contains cycles of $\lfloor\frac{k}{2}\rfloor$ distinct odd lengths. We strengthen this prominent result by showing that such $G$ contains cycles of $\lfloor\frac{k}{2}\rfloor$ consecutive odd lengths. Along the way, combining extremal and structural tools, we prove a stronger statement that every graph of chromatic number $k+1\geq 7$ contains $k$ cycles of consecutive lengths, except that some block is $K_{k+1}$. As corollaries, this confirms a conjecture of Verstraëte and answers…
Let $H =(\mathcal{M} \cup \mathcal{J} ,E \cup \mathcal{E})$ be a hypergraph with two hypervertices $\mathcal{G}_1$ and $\mathcal{G}_2$ where $\mathcal{M} =\mathcal{G}_{1} \cup \mathcal{G}_{2}$ and $\mathcal{G}_{1} \cap \mathcal{G}_{2} =\varnothing $. An edge $\{h ,j\} \in E$ in a bi-partite multigraph graph $(\mathcal{M} \cup \mathcal{J} ,E)$ has an integer multiplicity $b_{j h}$, and a hyperedge $\{\mathcal{G}_{\ell } ,j\} \in \mathcal{E}$, $\ell=1,2$, has an integer multiplicity $a_{j \ell }$. It has been conjectured in [5] that $χ\prime (H) =\lceil χ\prime _{f} (H)\rceil $, where $χ\prime (…
We study the class of simple graphs $\mathcal{G}^*$ for which every pair of distinct odd cycles intersect in at most one edge. We give a structural characterization of the graphs in $\mathcal{G}^*$ and prove that every $G \in \mathcal{G}^*$ satisfies the list-edge-coloring conjecture. When $Δ(G) \geq 4$, we in fact prove a stronger result about kernel-perfect orientations in $L(G)$ which implies that $G$ is $(mΔ(G):m)$-edge-choosable and $Δ(G)$-edge-paintable for every $m \geq 1$.
The Chromatic Symmetric Function for Unicyclic Graphs
Universidad de Chile（智利大学）；Stephen F. Austin State University（斯蒂芬·F·奥斯汀州立大学）；Dartmouth College（达特茅斯学院）；Arizona State University（亚利桑那州立大学）；Willamette University（威拉米特大学）
Scott proved in 1997 that for any tree $T$, every graph with bounded clique number which does not contain any subdivision of $T$ as an induced subgraph has bounded chromatic number. Scott also conjectured that the same should hold if $T$ is replaced by any graph $H$. Pawlik et al. recently constructed a family of triangle-free intersection graphs of segments in the plane with unbounded chromatic number (thereby disproving an old conjecture of Erdős). This shows that Scott's conjecture is false whenever $H$ is obtained from a non-planar graph by subdividing every edge at least once. It remains …
A variety of the mutual-visibility coloring problem for graphs
Cochin University of Science and Technology（科钦科技大学）；University of Maribor（马里博尔大学）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）；Universidad de Cádiz（加的斯大学）
More than ten years ago in 2008, a new kind of graph coloring appeared in graph theory, which is the {\it rainbow connection coloring} of graphs, and then followed by some other new concepts of graph colorings, such as {\it proper connection coloring, monochromatic connection coloring, and conflict-free connection coloring} of graphs. In about ten years of our consistent study, we found that these new concepts of graph colorings are actually quite different from the classic graph colorings. These {\it colored connection colorings} of graphs are brand-new colorings and they need to take care of…
