I was searching for graph coloring algorithms, and I have found algorithm, which, how author states, runs in polynomial time. Author gives also C++ program source code and demonstration program.
The suspicious thing is that decision problem whether graph is k-colorable, is NP-complete, so no polynomial time algorithm should exist until P=NP.
However, author doesn't claims, that algorithm works for all graphs, he only says, that he haven't found any graph, for which algorithm doesn't work.
So, the question: does that algorithm really works for every graph and that means actually P=NP, or there exist certain graphs/graph classes for which it doesn't work? Or maybe there is simply a mistake in complexity calculation?
I think you haven't read the abstract very carefully.
The author presents an algorithm which finds m-colorings of a graph, for some m less than the limit imposed by Brooks' theorem: https://en.wikipedia.org/wiki/Brooks'_theorem
(which is old and states that chi < delta + 1 as the author states in second sentence.)
The author is aware of the P vs NP question. The paper does not claim to resolve the question, he merely states:
For all known examples of graphs, the algorithm finds a proper m-coloring of the vertices of the graph G for m equal to the chromatic number χ(G)
Then he asks,
In view of the importance of the P versus NP question, we ask: does there exist a graph G for which this algorithm cannot find a proper m-coloring of the vertices of G with m equal to the chromatic number χ(G)?
Emphasis in original (!)
So it doesn't claim to resolve P vs NP, its just, as a matter of academic research, they ask "can anyone produce an example on which this algorithm fails to reach the chromatic number", which might be instructive to them for mathematical purposes. It is highly unlikely that the algorithm actually achieves the chromatic number for all graphs. (Although it is, scientifically speaking, unknown whether it does or doesn't.)