Cooperative coloring of some graph families

Abstract

In a family ${G_1, G_2, \ldots, G_m}$ of graphs sharing the same vertex set $V$, a cooperative coloring involves selecting one independent set $I_i$ from $G_i$ for each $i\in \{1,2,\ldots,m\}$ such that $\bigcup_{i=1}^m I_i = V$. For a graph class $\mathcal{G}$, let $m_{\mathcal{G}}(d)$ denote the minimum $m$ required to ensure that any graph family ${G_1, G_2, \ldots, G_m}$ on the same vertex set, where $G_i\in\mathcal{G}$ and $Δ(G_i)\leq d$ for each $i\in \{1,2,\ldots,m\}$, admits a cooperative coloring. For the graph classes $\mathcal{T}$ (trees) and $\mathcal{W}$ (wheels), we find that $m_\mathcal{T}(3)=4$ and $m_\mathcal{W}(4)=5$. Also, we prove that $m_{\mathcal{B}^*}(d)=O(\log_2 d)$ and $m_{\mathcal{L}}(d)=O\left(\frac{\log d}{\log\log d}\right)$, where $\mathcal{B}^*$ represents the class of graphs whose components are balanced complete bipartite graphs, and $\mathcal{L}$ represents the class of graphs whose components are generalized theta graphs.

Figures (5)

• Figure 1: In our construction illustrating that $m_\mathcal{T}(3)\geq 4$, these two graphs are elemental subgraphs. It is easy to check that $H_1$ does not admit an adapted coloring using color set $\{1,2\}$.
• Figure 2: This is a graph that has been edge-colored with three colors and does not admit an adapted coloring.
• Figure 3: In the graph depicted, each dotted line represents three edges that are colored differently. For instance, in $W_3$, every pair of vertices is connected by three edges, each assigned a distinct edge color from $\{1,2,3\}$. The graphs labeled as $H_1$, $H_2$, $H_3$, and $H_4$ are copies of $W_3$ with edge colorings that have been shifted.
• Figure 4: An edge-colored multigraph using four colors, where vertex $v$ is connected to each vertex in $H_i$ by an edge colored $i$ for each $1\leq i\leq 4$. This graph does not admit an adapted coloring.
• Figure 5: The first two graphs of the recursively constructed graphs $(G_t,\phi_t)$.

Theorems & Definitions (21)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1
• proof
• Theorem 2.2