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Register a new userStructural properties of edge-chromatic critical multigraphs
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Appearing in different format, Gupta,(1967), Goldberg,(1973), Andersen,(1977), and Seymour,(1979) conjectured that if $G$ is an edge-$k$-critical graph with $k ge Delta +1$, then $|V(G)|$ is odd and, for every edge $e$, $E(G-e)$ is a union of disjoint near-perfect matchings, where $Delta$ 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 Goldbergs conjecture. In particular, we showed that Goldbergs conjecture holds for graph $G$ with $|V(G)| le 39$ and $|Delta(G)| le 39$ and Jacobsens equivalent conjecture holds for $m le 39$ while the previous known bound is $23$.
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Given a graph $G$, denote by $Delta$, $bar{d}$ and $chi^prime$ the maximum degree, the average degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {it edge-$Delta$-critical} if $chi^prime(G)=Delta+1$ and $chi^prime(H)leDelta$ for every proper subgraph $H$ of $G$. Vizing in 1968 conjectured that if $G$ is edge-$Delta$-critical, then $bar{d}geq Delta-1+ frac{3}{n}$. We show that $$ begin{displaystyle} avd ge begin{cases} 0.69241D-0.15658 quad,: mbox{ if } Deltageq 66, 0.69392D-0.20642quad;,mbox{ if } Delta=65, mbox{ and } 0.68706D+0.19815quad! quadmbox{if } 56leq Deltaleq64. end{cases} end{displaystyle} $$ This result improves the best known bound $frac{2}{3}(Delta +2)$ obtained by Woodall in 2007 for $Delta geq 56$. Additionally, Woodall constructed an infinite family of graphs showing his result cannot be improved by well-known Vizings Adjacency Lemma and other known edge-coloring techniques. To over come the barrier, we follow the recently developed recoloring technique of Tashkinov trees to expand Vizing fans technique to a larger class of trees.
Let $G$ be a simple graph with maximum degree $Delta(G)$ and chromatic index $chi(G)$. A classic result of Vizing indicates that either $chi(G )=Delta(G)$ or $chi(G )=Delta(G)+1$. The graph $G$ is called $Delta$-critical if $G$ is connected, $chi(G )=Delta(G)+1$ and for any $ein E(G)$, $chi(G-e)=Delta(G)$. Let $G$ be an $n$-vertex $Delta$-critical graph. Vizing conjectured that $alpha(G)$, the independence number of $G$, is at most $frac{n}{2}$. The current best result on this conjecture, shown by Woodall, is that $alpha(G)<frac{3n}{5}$. We show that for any given $varepsilonin (0,1)$, there exist positive constants $d_0(varepsilon)$ and $D_0(varepsilon)$ such that if $G$ is an $n$-vertex $Delta$-critical graph with minimum degree at least $d_0$ and maximum degree at least $D_0$, then $alpha(G)<(frac{{1}}{2}+varepsilon)n$. In particular, we show that if $G$ is an $n$-vertex $Delta$-critical graph with minimum degree at least $d$ and $Delta(G)ge (d+2)^{5d+10}$, then [ alpha(G) < left. begin{cases} frac{7n}{12}, & text{if $d= 3$; } frac{4n}{7}, & text{if $d= 4$; } frac{d+2+sqrt[3]{(d-1)d}}{2d+4+sqrt[3]{(d-1)d}}n<frac{4n}{7}, & text{if $dge 19$. } end{cases} right. ]
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 $chi(G)$ (resp. $chi^*(G)$). Let $Delta(G)$ be the maximum degree of $G$ and let [Gamma(G)=max Big{frac{2|E(U)|}{|U|-1}:,, U subseteq V, ,, |U|ge 3 hskip 2mm {rm and hskip 2mm odd} Big},] where $E(U)$ is the set of all edges of $G$ with both ends in $U$. Clearly, $max{Delta(G), , lceil Gamma(G) rceil }$ is a lower bound for $chi(G)$. As shown by Seymour, $chi^*(G)=max{Delta(G), , Gamma(G)}$. In the 1970s Goldberg and Seymour independently conjectured that $chi(G) le max{Delta(G)+1, , lceil Gamma(G) rceil}$. Over the past four decades this conjecture, a cornerstone in modern edge-coloring, has been a subject of extensive research, and has stimulated a significant body of work. In this paper we present a proof of this conjecture. Our result implies that, first, there are only two possible values for $chi(G)$, so an analogue to Vizings theorem on edge-colorings of simple graphs, a fundamental result in graph theory, holds for multigraphs; second, although it is $NP$-hard in general to determine $chi(G)$, we can approximate it within one of its true value, and find it exactly in polynomial time when $Gamma(G)>Delta(G)$; third, every multigraph $G$ satisfies $chi(G)-chi^*(G) le 1$, so FECP has a fascinating integer rounding property.
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 $P(H,lambda)$ is the chromatic polynomial of a hypergraph $H$ which is constructed from $G$. Finally we show that the multiplicity of root $0$ of $P(H,lambda)$ may be at least $2$ for some connected hypergraphs $H$, and the multiplicity of root $1$ of $P(H,lambda)$ may be $1$ for some connected and separable hypergraphs $H$ and may be $2$ for some connected and non-separable hypergraphs $H$.
In this note we obtain a new bound for the acyclic edge chromatic number $a(G)$ of a graph $G$ with maximum degree $D$ proving that $a(G)leq 3.569(D-1)$. To get this result we revisit and slightly modify the method described in [Giotis, Kirousis, Psaromiligkos and Thilikos, Theoretical Computer Science, 66: 40-50, 2017].
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