Coloring of the incidentors and vertices of an undirected multigraph (Q2760693)

Let \(G=(V,E)\) be a undirected loopless multigraph. If an edge \(e\) is incident with a vertex \(v\), then the pair \((v,e)\) is called an incidentor attached to \(v\). An incidentor coloring of \(G\) with integers is \(p\)-admissible if the colors of every two adjacent incidentors, i.e., having a vertex in common, differ by at least \(p\). A total \(p\)-coloring of \(G\) is a coloring of the vertices and incidentors such that every two adjacent vertices are colored differently, the incidentors are colored \(p\)-admissibly, and the color of each vertex \(v\) differs from the color of each incidentor attached to \(v\). The authors present a precise formula for the incidentor \(p\)-chromatic number \(\chi I(p,G)\) for any integer \(p\) and multigraph \(G\). For the total \(p\)-chromatic number \(\tau(p,G)\) it is proven that \(\chi I(p,G)\leq\tau(p,G)\leq\chi I(p,G)+1\).