@article{bc5bd4dce4174ea2803b9568f168b110,

title = "The total chromatic number of any multigraph with maximum degree five is at most seven",

abstract = "The result announced in the title is proved. A new proof of the total 6-colorability of any multigraph with maximum degree 4 is also given.",

author = "Kostochka, {A. V.}",

note = "Funding Information: The validity of (1) is known to be true for graphs in several wide families (see \[3, 4\]). Hind \[6\]a nd then Chetwynd and H~iggkvist \[5\]p roved that it is 'almost true', i.e. that z2(G) ~< A + o(A) for each simple graph G with maximum degree A. But the exact bound (1) was published only for A ~< 3 \[14, 15\] and for A = 4 \[7\].T he inequality (1) is not true if we replace simple graphs by multigraphs, since for each A ~> 1, there exists a multigraph S (so-called Shannon's triangle) with maximum degree A and zI(G) = L1.5A J. In \[10\]( cf. also \[8\])i t was proved that for any integer A >~ 4, A ~ 5 * This research described in this publication was made possible in part by Grant N RPY000 from the International Science Foundation. This work was also partly supported by Grant 93-01-01486 of the Russian Foundation of Fundamental Research.",

year = "1996",

month = dec,

day = "25",

doi = "10.1016/0012-365X(95)00286-6",

language = "English (US)",

volume = "162",

pages = "199--214",

journal = "Discrete Mathematics",

issn = "0012-365X",

publisher = "Elsevier",

number = "1-3",

}