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Critical star multigraphs. (English) Zbl 0619.05023
Authors’ abstract: ”A star-multigraph G is a multigraph in which there is a vertex $$v^*$$ which is incident with each non-simple edge. It is critical if it is connected, class 2 and $$\chi '(G\setminus e)<\chi '(G)$$ for each $$e\in E(G)$$. We show that, if G is any star multigraph, then $$\chi '(G)\leq \Delta (G)+1$$. We investigate the edge-chromatic class of star multigraphs with at most two vertices of maximum degree. We also obtain a number of results on critical star multigraphs. We shall make use of these results in later papers.”
Reviewer: R.L.Hemminger

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C35 Extremal problems in graph theory
##### Keywords:
chromatic index; star graphs; multigraph
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##### References:
 [1] Andersen, L.D.: On edge-colourings of graphs. Math. Scand.,40, 161–175 (1977) · Zbl 0373.05035 [2] Behzad, M., Chartrand, G.: Introduction to the Theory of Graphs. Allyn and Bacon 1971 · Zbl 0238.05101 [3] Bollobás, B.: Extremal Graph Theory. London: Academic Press, 1978 · Zbl 1099.05044 [4] Chetwynd, A.G., Hilton, A.J.W.: Partial edge-colourings of complete graphs or of graphs which are nearly complete. In: Graph Theory and Combinatorics (Proc. of the Cambridge Combinatorial Conference in Honour of Paul Erdös, edited by B. Bollobás), pp. 81–98. London: Academic Press 1984 · Zbl 0549.05027 [5] Chetwynd, A.G., Hilton, A.J.W.: The chromatic index of graphs of even order with many edges. J. Graph Theory,8, 463–470 (1984) · Zbl 0562.05024 [6] Chetwynd, A.G., Hilton, A.J.W.: Regular graphs of high degree are 1-factorizable. Proc. London Math. Soc.,50, 193–206 (1985) · Zbl 0561.05027 [7] Chetwynd, A.G., Hilton, A.J.W.: The chromatic class of graphs with at most four vertices of maximum degree. Congr. Numerantium43 (Proc. of the 15th Southeastern Conference on Graph Theory Combinatorics and Computing) 221–248 (1984) · Zbl 0561.05023 [8] Chetwynd, A.G., Hilton, A.J.W.: Star multigraphs with three vertices of maximum degree. Math. Proc. Camb. Philos. Soc. (to appear) · Zbl 0716.05021 [9] Chetwynd, A.G., Hilton, A.J.W.: The edge-chromatic class of graphs with large maximum degree, where the number of vertices of maximum degree is relatively small (submitted) · Zbl 0716.05021 [10] Chetwynd, A.G., Hilton, A.J.W.: The edge-chromatic class of graphs with maximum degree at least |V| 3 (submitted) · Zbl 0692.05032 [11] Chetwynd, A.G., Hilton, A.J.W.: The edge-chromatic class of graphs of even order with maximum degree at least |V| 4 (in preparation) · Zbl 0692.05032 [12] Fiorini, S., Wilson, R.J.: Edge-colourings of graphs, Res. Notes Math.16 (1977) · Zbl 0421.05023 [13] Gol’dberg, M.K.: Remark on the chromatic class of a multigraph (in Russian). Vyčisl. Mat. i Vyčisl. Tehn. (Kharkov)5, 128–130 (1974) [14] Hilton, A.J.W.: Definitions of criticality with respect to edge-colouring. J. Graph Theory,1, 61–68 (1977) · Zbl 0373.05054 [15] Hilton, A.J.W., Jackson, Bill: A note concerning the chromatic index of multigraphs. J. Graph Theory (to appear) · Zbl 0651.05033 [16] Hilton, A.J.W., Johnson, P.D.: Graphs which are critical with respect to the chromatic index (submitted) · Zbl 0725.05041 [17] Hilton, A.J.W., Rodger, C.A.: Triangulating nearly complete graphs of odd order (in preparation) [18] Plantholt, M.: The chromatic index of graphs with a spanning star. J. Graph Theory5, 5–13 (1981) · Zbl 0448.05031 [19] Vizing, V.G.: On an estimate of the chromatic class of a p-graph (in Russian). Diskret. Analiz.3, 25–30 (1964) [20] Vizing, V.G.: Critical graphs with a given chromatic class (in Russian). Diskret. Analiz.5, 9–17 (1965)
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