On total irregular labelings with no-hole weights of some planar graphs (Q6599821)

Given a graph \(G=(V,E)\), an assignment \(\partial\) of positive integers \(1,2,\dots,k\) to the vertices and edges of \(G\) is called an edge (vertex) irregular total \(k\)-labeling of \(G\) if all edge (vertex) weights are distinct, where the edge is given by \(\varphi(e=uv)=\partial(u)+\partial(e)+\partial(v)\), while the vertex weight is defined as \(\varphi(v)=\partial(v)+\sum_{uv\in E}\partial(uv)\). The total edge (vertex) irregularity strength of \(G\) is the smallest \(k\) such that \(G\) has an edge (vertex) irregular total \(k\)-labeling and denoted by \(\mathrm{tes}(G)\) and \(\mathrm{tvs}(G)\), respectively. Moreover, the minimal \(k\) for which \(G\) admits an edge (vertex) irregular total \(k\)-labeling with no-hole weights, i.e., the edge (vertex) weights constitute a set of consecutive integers, is denoted by \(\gamma(G)\) and \(\gamma'(G)\), respectively.\N\NIn this paper, the exact values of \(\gamma(G)\) are determined for several graph classes, namely the bistar \(B_{n,n}\) for \(n\ge 1\), the triangular snake \(TS_{2n+1}\) for \(n\ge 1\), the double triangular snake \(DTS_{3n+1}\) for \(n\ge 1\), the quadrilateral snake \(QSk_{3n+1}\) for \(n\ge 1\), and the corona product \(C_n\odot\overline{K_m}\) for \(n\ge 2\) and \(m\ge 1\). Additionally, it is shown that \(\gamma'(B_{n,n})=2n+3-m\), where \(m\) is the smallest positive integer such that \(n\le\lfloor(m^2+m)/4\rfloor-1\).