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1)I try to show that the graphs with total chromatic number $\chi''(G)=\chi'(G)+\chi(G)$ are exactly the bipartite graphs.

Definition https://en.wikipedia.org/wiki/Total_coloring

After the answer of Smylic

2)If $\chi''(G)=\chi'(G)+\chi(G)$ holds then the graph should be bipartite

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    Now posted to MO, http://mathoverflow.net/questions/262879/total-chromatic-number-and-bipartite-graphs – 2017-02-22

1 Answers 1

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This is not true, because $\chi''(C_6) = 3 \ne 4 = 2 + 2 = \chi'(C_6) + \chi(C_6)$.

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    i will edit it. the right seems to be that if this holds the graph should be bipartite – 2017-02-13
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    I would like to give you another counterexample, but it should be strong enough to break Behzad and Vizing conjecture, that $\chi''(G) \le \Delta(G) + 2$, because $\chi(G) \ge 3$ for non-bipartite graph $G$ and $\chi'(G) \ge \Delta(G)$. – 2017-02-13
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    Of course but what i ask is far away of being so strong..... – 2017-02-14
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    I strongly believe that $\chi''(C_6) = 4$ – 2017-02-20
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    All right, let $c\colon V \cup E \to \mathbb{N}_0$ be a color function. Possible coloring is $c(v_i) = i \bmod 3$ for $1 \le i \le 6$ and $c(\{\,v_i, v_{i + 1}\,\}) = i + 2$ for $0 \le i \le 5$ assuming $v_0 = v_6$. – 2017-02-20
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    yes,got it....im waiting for an answer to the second .... – 2017-02-20
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    I guess it is rather hard research question, more suitable on mathoverflow. – 2017-02-20
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    ok thnxi will make atry – 2017-02-20