MHB Graph two colours, no monochromatic path.

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In graph theory, the problem involves coloring the vertices of a graph G, where all vertices have degrees of 3 or less, using two colors to ensure there are no monochromatic paths of length 3. A suggested approach is to analyze the structure of the graph and the connections of vertices to avoid creating such paths. Additionally, a lemma indicates that if all vertices have degrees greater than or equal to d, there must exist a path of at least length d within the graph. This relationship between vertex degree and path length is crucial for understanding graph properties. The discussion emphasizes the need for practical solutions and insights into these concepts.
Arnold1
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I've just begun studying graph theory and I have some difficulty with this problem. Could you tell me how to go about solving it? I would really appreciate the least formal solution possible.

In a graph G all vertices have degrees \le 3. Show that we can color its vertices in two colors so that in G there exists no one-color path, whose length is 3.

And a similar one.
There's this quite popular lemma that if in a graph all vertices have degrees \ge d, then in this graph there's a path whose length is d.
 
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Arnold said:
There's this quite popular lemma that if in a graph all vertices have degrees \ge d, then in this graph there's a path whose length is d.

Let v_0 v_1 ... v_k be a path of maximum length in a graph G. Then all neighbours of v_0 lie on the path. Since \deg (v_0) \geq \delta (G), we have k \geq \delta (G) and the path has at least length \delta (G).
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...

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