Mathematics

CLC number: O 157.5

Gracefulness of Two Kinds of Unconnected Graphs with Even Vertices

College of Science, North China Institute of Science and Technology, Sanhe 065201, Hebei, China

Received: 10 February 2022

Abstract

Two kinds of unconnected double fan graphs with even vertices,$(P_{\mathrm{1}}^{(\mathrm{1})}\vee (P_{\mathrm{2}n}^{(\mathrm{1})}\bigcup P_{\mathrm{2}n}^{(\mathrm{2})}))\bigcup P_{\mathrm{2}n+\mathrm{1}}\bigcup (P_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{K_{\mathrm{2}n}})$ and $(P_{\mathrm{1}}^{(\mathrm{1})}\vee (P_{\mathrm{2}n}^{(\mathrm{1})}\bigcup$ $P_{\mathrm{2}n}^{(\mathrm{2})}))\bigcup (P_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{K_{\mathrm{2}n}^{(\mathrm{1})}})\bigcup (P_{\mathrm{1}}^{(\mathrm{3})}\vee \overline{K_{\mathrm{2}n}^{(\mathrm{2})}})$ were presented. For natural number $n\in \mathbf{N}$,$n\ge \mathrm{1}$,the two graphs are all graceful graphs, where $P_{\mathrm{2}n}^{(\mathrm{1})},P_{\mathrm{2}n}^{(\mathrm{2})}$ are even-vertices path,$P_{\mathrm{2}n+\mathrm{1}}$ is odd-vertices path,$\overline{K_{\mathrm{2}n}},\overline{K_{\mathrm{2}n}^{(\mathrm{1})},}\overline{K_{\mathrm{2}n}^{(\mathrm{2})}}$ are the complement of graph ${K_{\mathrm{2}}}_n$,$G_{\mathrm{1}}\vee G_{\mathrm{2}}$ is the join graph of $G_{\mathrm{1}}$ and $G_{\mathrm{2}}$.