Edge coloring of products of signed graphs

Abstract

In 2020, Behr defined the problem of edge coloring of signed graphs and showed that every signed graph $(G, σ)$ can be colored using exactly $Δ(G)$ or $Δ(G) + 1$ colors, where $Δ(G)$ is the maximum degree in graph $G$. In this paper, we focus on products of signed graphs. We recall the definitions of the Cartesian, tensor, strong, and corona products of signed graphs and prove results for them. In particular, we show that $(1)$ the Cartesian product of $Δ$-edge-colorable signed graphs is $Δ$-edge-colorable, $(2)$ the tensor product of a $Δ$-edge-colorable signed graph and a signed tree requires only $Δ$ colors and $(3)$ the corona product of almost any two signed graphs is $Δ$-edge-colorable. We also prove some results related to the coloring of products of signed paths and cycles.

Figures (3)

• Figure 1: Example signed graphs considered in \ref{['cartesian_path_cycle']} -- $(a)$$(P_3 \square C_4, \sigma)$, $(b)$$(P_4 \square C_4, \sigma)$. Edges belonging to graph $H_1$ are marked with bold lines in both cases.
• Figure 2: Example graph $(C_6 \square C_4, \sigma)$ considered in \ref{['cartesian_c2r_c2s']}. Edges of respective graphs are marked with solid lines.
• Figure 4: An example corona product $P_4 \odot K_3$. Edges of $P_4$ and copies of $K_3$ are marked with solid lines. Edges connecting them are marked with dashed lines.

Theorems & Definitions (26)

• Theorem 1: behr
• Theorem 2: behr
• Theorem 3: behr
• Theorem 4: classes_one_two
• Corollary 5
• Theorem 6
• Corollary 7
• Theorem 8
• Lemma 9
• Lemma 10: classes_one_two