Lines unfortunately cover node descriptions (tikz)

Question

I'd like to visualize the KS-type proof by Mermin on p. 17 in arXiv:1802.10119 by using colored thick lines below the node descriptions and not on the descriptions. But in my code, the lines cover the descriptions and I don't know how to fix it. Additionally it would be nice if the lines would not stop before the node description, such that the pentagram can be seen completely. I'm not very familiar with tikz, therefore I positioned the nodes manually, which is okay - but I hope someone could help me with the coloring problem and making this picture look neater.

The code so far:

\begin{tikzpicture}
\node(A)[draw=none,fill=none] at (0,0.5) {$\sigma^1_y$};
\node(B)[draw=none,fill=none] at (-0.8,-1){$\sigma_y^1\sigma_x^2\sigma_y^3$};
\node(C)[draw=none,fill=none] at (0.8,-1){$\sigma_y^1\sigma_y^2\sigma_x^3$};
\node(D)[draw=none,fill=none] at (2.4,-1){$\sigma_x^1\sigma_y^2\sigma_y^3$};
\node(E)[draw=none,fill=none] at (-2.4,-1){$\sigma_x^1\sigma_x^2\sigma_x^3$};
\node(F)[draw=none,fill=none] at (1.1,-1.9){$\sigma_y^3$};
\node(G)[draw=none,fill=none] at (-1.1,-1.9){$\sigma_x^3$};
\node(H)[draw=none,fill=none] at (0,-2.6){$\sigma_x^1$};
\node(I)[draw=none,fill=none] at (-2,-4){$\sigma_y^2$};
\node(J)[draw=none,fill=none] at (2,-4){$\sigma_x^2$};

\draw[magenta,thick](A)--(J);
\draw[magenta,thick](A)--(I);
\draw[magenta,thick](E)--(J);
\draw[magenta,thick](D)--(I);
\draw[blue,thick](D)--(E);
\end{tikzpicture}

Answer 1

If you want to improve the node positions you can also use the intersections library and the backgrounds library to make sure the lines are in the background:

\documentclass[tikz]{standalone}
\usetikzlibrary{backgrounds,calc,intersections}

\begin{document}
\begin{tikzpicture}
\coordinate (A) at (0,0.5);
\coordinate (J) at (2,-4);
\coordinate (E) at (-2.4,-1);
\coordinate (D)at (2.4,-1);
\coordinate (I) at (-2,-4);

\path[name path = de] (D)--(E);
\path[name path=aj] (A) -- (J);
\path[name path = ai] (A) -- (I);
\path[name path =di] (D) --(I) ;
\path[name path =ej] (E) --(J);
\path[name intersections ={of = aj and de,by=B}] ;
\path[name intersections ={of = ai and de,by=C}] ;
\path[name intersections = {of=aj and di,by=F}];
\path[name intersections = {of =ai and ej,by=G}];
\path[name intersections = {of =di and ej,by=H}];

\node at (C) {C};
\node at (B) {B};
\node at (F){F};
\node at (G){G};
\node at (H){H};

\begin{pgfonlayer}{background}
\draw[magenta,thick] (I) -- (A) node[black] {$\sigma^1_y$}--(J) node [black]{J}-- (E) node [black]{E} -- (D) node [black]{D} -- (I) node [black]{I};
\draw[blue,thick](D)--(E);
\end{pgfonlayer}
\end{tikzpicture}
\end{document}

(EDIT to the image: replaced the image with an image using the acutal colors.)

EDIT: Now that I can properly do this at a stationary device, I removed some of the redudancies of the code and used polar coordinates to ensure a regular pentagramm. Also, I changed node labels to be more mathematical.

\documentclass[tikz]{standalone}
\usetikzlibrary{backgrounds,calc,intersections}

\begin{document}
    \begin{tikzpicture}
        \foreach \i/\j in {0/A, 1/B, 2/C, 3/D, 4/E}{
            \coordinate (\j) at ({90+72*\i}:2cm);
            }
        
        \path[name path = ac] (A) -- (C);
        \path[name path = ad] (A) -- (D);
        \path[name path = be] (B) -- (E);
        \path[name path = bd] (B) -- (D);
        \path[name path = ce] (E) -- (C);
        
        \node[name intersections = {of = ad and be,by=F}] at (F) {F};
        \node[name intersections = {of = ac and be,by=G}] at (G) {G};
        \node[name intersections = {of = ac and bd,by=H}] at (H) {H};
        \node[name intersections = {of = ce and bd,by=I}] at (I) {I};
        \node[name intersections = {of = ad and ce,by=J}] at (J) {J};
        
        \begin{pgfonlayer}{background}
            \begin{scope}[every node/.style={black}]
                \draw[magenta,thick] (A) node {A} -- (C) node {C} -- (E) node {E} -- (B) node {B} -- (D) node {D} -- cycle;
                \draw[blue,thick] (B)--(E);
            \end{scope}
        \end{pgfonlayer}
    \end{tikzpicture}
\end{document}

I should point out that it is of course not very nice having to draw the blue line again, on top of the magenta line, but unfortunately that is necessary to obtain the pointy edges. Changing the color for only a section of such a path is a bit more complicated and has been answered in great detail for this question: TikZ: changing colour of a path half way along

Answer 2

With this:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{intersections}
\begin{document}
\begin{tikzpicture}[scale=4]
\foreach \x in{0,...,4} \coordinate (A\x) at ({sin(\x*72)},{cos(\x*72)});
\draw[name path=A02,thick,color=magenta] (A0)--(A2);
\draw[name path=A03,thick,color=magenta] (A0)--(A3);
\draw[thick,color=magenta] (A4)--(A2);
\draw[thick,color=magenta] (A3)--(A1);
\draw[name path=A14,thick,blue] (A4)--(A1);
\path[name intersections={of=A02 and A14,by=Ir}];
\path[name intersections={of=A03 and A14,by=Il}];
\node[above] at (A0) {$\sigma_y^1$};
\node[right] at (A1) {$\sigma_x^1\sigma_y^2\sigma_y^3$};
\node[below right] at (A2) {$\sigma_x^2$};
\node[below left] at (A3) {$\sigma_y^2$};
\node[left] at (A4) {$\sigma_x^1\sigma_x^2\sigma_x^3$};
\node[above right] at (Ir) {$\sigma_y^1\sigma_x^2\sigma_y^3$};
\node[above left] at (Il) {$\sigma_y^1\sigma_y^2\sigma_x^3$};
\end{tikzpicture}
\end{document}

I got this:

Answer 3

Now it's beautiful, at least I'm happy with the result. I added white filling for better readability.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{intersections}
\usetikzlibrary{backgrounds,calc,intersections}

\begin{document}
\begin{tikzpicture}[scale=4]
\foreach \x in{0,...,4} \coordinate (A\x) at ({sin(\x*72)},{cos(\x*72)});
\path[name path = a1] (A0)--(A2);
\path[name path=a2] (A0) -- (A3);
\path[name path = a3] (A4) -- (A2);
\path[name path =a4] (A3) --(A1) ;
\path[name path =a5] (A4) --(A1);
\path[name intersections ={of = a2 and a3,by=B1}] ;
\path[name intersections ={of = a5 and a1,by=B2}] ;
\path[name intersections = {of=a3 and a4,by=B3}];
\path[name intersections = {of =a1 and a4,by=B4}];
\path[name intersections = {of =a5 and a2,by=B5}];


\node [black,rectangle,fill=white] at (B1) {$\sigma_x^3$};
\node [black,rectangle,fill=white] at (B2) {$\sigma_y^1\sigma_x^2\sigma_y^3$};
\node [black,rectangle,fill=white] at (B3){$\sigma_x^1$};
\node [black,rectangle,fill=white] at (B4){$\sigma_y^3$};
\node [black,rectangle,fill=white] at (B5){$\sigma_y^1\sigma_y^2\sigma_x^3$};

\begin{pgfonlayer}{background}
\draw[magenta,line width=1mm] (A0) -- (A2) node[black,rectangle,fill=white] {$\sigma_x^2$}--(A4) -- (A1)  -- (A3) node [black,rectangle,fill=white]{$\sigma_y^2$} -- (A0) node [black,rectangle,fill=white]{$\sigma_y^1$};
\draw[blue,line width=1mm](A4)node [black,rectangle,fill=white]{$\sigma_x^1\sigma_x^2\sigma_x^3$}--(A1)node [black,rectangle,fill=white]{$\sigma_x^1\sigma_y^2\sigma_y^3$};
\end{pgfonlayer}

\end{tikzpicture}
\end{document}