Chromatic‐index critical multigraphs of order 20

A multigraph M with maximum degree Δ(M) is called critical, if the chromatic index χ′(M) > Δ(M) and χ′(M − e) = χ′(M) − 1 for each edge e of M. The weak critical graph conjecture [1, 7] claims that there exists a constant c > 0 such that every critical multigraph M with at most c · Δ(M) vertices has odd order. We disprove this conjecture by constructing critical multigraphs of order 20 with maximum degree k for all k ≥ 5. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 240–245, 2000     

Edge-coloring critical graphs with high degree

It is proved here that any edge-coloring critical graph of order n and maximum degree Δ?8 has the size at least 3(n+Δ−8). It generalizes a result of Hugh Hind and Yue Zhao.     

A Note on Adjacent Strong Edge Coloring of K（n,m）

In this paper, we prove that the adjacent strong edge chromatic number of a graph K（n,m） is n ＋ 1, with n ≥ 2, m ≥ 1.     

On fans in multigraphs

We introduce a unifying framework for studying edge‐coloring problems on multigraphs. This is defined in terms of a rooted directed multigraph , which is naturally associated to the set of fans based at a given vertex u in a multigraph G. We call the “Fan Digraph.” We show that fans in G based at u are in one‐to‐one correspondence with directed trails in starting at the root of . We state and prove a central theorem about the fan digraph, which embodies many edge‐coloring results and expresses them at a higher level of abstraction. Using this result, we derive short proofs of classical theorems. We conclude with a new, generalized version of Vizing's Adjacency Lemma for multigraphs, which is stronger than all those known to the author. © 2005 Wiley Periodicals, Inc. J Graph Theory 51: 301–318, 2006     

A note on minimal matching covered graphs

A graph is called matching covered if for its every edge there is a maximum matching containing it. It is shown that minimal matching covered graphs without isolated vertices contain a perfect matching.     

On the critical point‐arboricity graphs

In this paper, we study the critical point‐arboricity graphs. We prove two lower bounds for the number of edges of k‐critical point‐arboricity graphs. A theorem of Kronk is extended by proving that the point‐arboricity of a graph G embedded on a surface S with Euler genus g = 2, 5, 6 or g ≥ 10 is at most with equality holding iff G contains either K_2k?1 or K_2k?4 + C₅ as a subgraph. It is also proved that locally planar graphs have point‐arboricity ≤ 3 and that triangle‐free locally planar‐graphs have point‐arboricity ≤ 2. © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 50–61, 2002     

Edge Coloring of Embedded Graphs with Large Girth

Let G be a simple graph embedded in the surface of Euler characteristic ()0. Denote _e(G), and g the edge chromatic number, the maximum degree and the girth of the graph G, respectively. The paper shows that _e(G)= if 5 and g4, or 4 and g5, or 3 and g9. In addition, if ()>0, then _e(G)= if 3 and g8. Acknowledgments.The authors would like to thank Dr. C.Q. Zhang for carefully reading several versions of this paper during its preparation and for suggesting several stylistic changes that have improved the overall presentation.     

Erratum: The average degree of an edge-chromatic critical graph II

In the article “The average degree of an edge-chromatic critical graph II” by Douglas R. Woodall (J. Graph Theory 56 (2007), 194-218), it was claimed that the average degree of an edge-chromatic critical graph with maximum degree $\mathrm{\Delta }$ is at least $\text{◂⋅▸}\frac{2}{3}\left(\mathrm{\Delta }+1\right)$ if $\mathrm{\Delta }⩾2$, at least $\text{◂+▸}\frac{2}{3}\mathrm{\Delta }+1$ if $\mathrm{\Delta }⩾8$, and at least $\text{◂⋅▸}\frac{2}{3}\left(\mathrm{\Delta }+2\right)$ if $\mathrm{\Delta }⩾15$. Unfortunately there were mistakes in the proof of the last two of these results, which are now proved only if $\mathrm{\Delta }⩾18$ and $\mathrm{\Delta }⩾30$, respectively.     

Characterizations of competition multigraphs

The notion of a competition multigraph was introduced by C. A. Anderson, K. F. Jones, J. R. Lundgren, and T. A. McKee [C. A. Anderson, K. F. Jones, J. R. Lundgren, and T. A. McKee: Competition multigraphs and the multicompetition number, Ars Combinatoria 29B (1990) 185-192] as a generalization of the competition graphs of digraphs.In this note, we give a characterization of competition multigraphs of arbitrary digraphs and a characterization of competition multigraphs of loopless digraphs. Moreover, we characterize multigraphs whose multicompetition numbers are at most m, where m is a given nonnegative integer and give characterizations of competition multihypergraphs.     

Exhaustive generation of k‐critical ‐free graphs

We describe an algorithm for generating all k‐critical ‐free graphs, based on a method of Hoàng et al. (A graph G is k‐critical H‐free if G is H‐free, k‐chromatic, and every H‐free proper subgraph of G is ‐colorable). Using this algorithm, we prove that there are only finitely many 4‐critical ‐free graphs, for both and . We also show that there are only finitely many 4‐critical ‐free graphs. For each 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. In addition, we prove a number of characterizations for 4‐critical H‐free graphs when H is disconnected. Moreover, we prove that for every t, the class of 4‐critical planar ‐free graphs is finite. We also determine all 52 4‐critical planar P₇‐free graphs. We also prove that every P₁₁‐free graph of girth at least five is 3‐colorable, and show that this is best possible by determining the smallest 4‐chromatic P₁₂‐free graph of girth at least five. Moreover, we show that every P₁₄‐free graph of girth at least six and every P₁₇‐free graph of girth at least seven is 3‐colorable. This strengthens results of Golovach et al.     

Partitioning series-parallel multigraphs into υ*-excluding edge covers

We prove that, for any given vertex υ* in a series-parallel graph G, its edge set can be partitioned into k = min{κ'(G) + 1,δ(G)} subsets such that each subset covers all the vertices of G possibly except for υ*, where δ(G) is the minimum degree of G and κ'(G) is the edge-connectivity of G. In addition, we show that the results in this paper are best possible and a polynomial time algorithm can be obtained for actually finding such a partition by our proof.     

图 P2×Cn的均匀邻强边色数

对图G(V,E),一正常边染色f若满足(1)对(V)uv∈E(G),f[u]≠f[v],其中f[u]={f(uv)|uv∈E};(2)对任意i≠j,有||E|-|Ej||≤1,其中Ei={e| e∈E(G)且f(e)=i}.则称f为G(V,E)的一k-均匀邻强边染色,简称k-EASC,并且称Xcas(G)=min{k|存在G(V,E)的一k-EASC为G(V,E)的均匀邻强边色数.本文得到了图P2×Cn的均匀邻强边色数.     

Subcubic planar graphs of girth 7 are class I

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.     

A note on Vizing's independence number conjecture of edge chromatic critical graphs

In 1968, Vizing conjectured that, if G is a Δ-critical graph with n vertices, then α(G)?n/2, where α(G) is the independence number of G. In this note, we verify this conjecture for n?2Δ.     

Super Restricted Edge Connectivity of Regular Graphs

An edge cut of a connected graph is called restricted if it separates this graph into components each having order at least 2; a graph G is super restricted edge connected if G−S contains an isolated edge for every minimum restricted edge cut S of G. It is proved in this paper that k-regular connected graph G is super restricted edge connected if k > |V(G)|/2+1. The lower bound on k is exemplified to be sharp to some extent. With this observation, we determined the number of edge cuts of size at most 2k−2 of these graphs. Supported by NNSF of China (10271105); Ministry of Science and Technology of Fujian (2003J036); Education Ministry of Fujian (JA03147)     

Size of Monochromatic Double Stars in Edge Colorings

We show that in every r-coloring of the edges of K _n there is a monochromatic double star with at least \(\frac{n(r+1)+r-1}{r^2}\) vertices. This result is sharp in asymptotic for r = 2 and for r≥ 3 improves a bound of Mubayi for the largest monochromatic subgraph of diameter at most three. When r-colorings are replaced by local r-colorings, our bound is \(\frac{n(r+1)+r-1}{r^2+1}\).     

On list critical graphs

In this paper we discuss some basic properties of k-list critical graphs. A graph G is k-list critical if there exists a list assignment L for G with |L(v)|=k−1 for all vertices v of G such that every proper subgraph of G is L-colorable, but G itself is not L-colorable. This generalizes the usual definition of a k-chromatic critical graph, where L(v)={1,…,k−1} for all vertices v of G. While the investigation of k-critical graphs is a well established part of coloring theory, not much is known about k-list critical graphs. Several unexpected phenomena occur, for instance a k-list critical graph may contain another one as a proper induced subgraph, with the same value of k. We also show that, for all 2≤p≤k, there is a minimal k-list critical graph with chromatic number p. Furthermore, we discuss the question, for which values of k and n is the complete graph K_nk-list critical. While this is the case for all 5≤k≤n, K_n is not 4-list critical if n is large.