Infinite families of planar graphs of a given injective chromatic number

TL;DR

This work investigates the injective chromatic number $\chi_i(G)$ of planar graphs, focusing on the conjectures of Chen et al. and Lužar–Škrekovski (and related girth-restricted variants). It employs an exact backtracking algorithm to compute $\chi_i(G)$ and to generate extensive computational evidence, confirming Lužar–Škrekovski's refined conjecture up to substantial sizes and producing infinite 3-connected planar families attaining the refined bounds for all relevant $\Delta$ (with $\Delta=4$ yielding a counterexample to Chen's bound but not sharp for the refined bound). The paper also constructs infinite families of 3-connected planar graphs meeting these bounds under higher girth constraints, and provides analogous results for related conjectures, supported by computational verification up to targeted orders. Overall, it significantly expands the landscape of infinite counterexamples to Chen's conjecture and sharpens our understanding of $\chi_i(G)$ in planar graphs across degrees and girths.

Abstract

An injective colouring of a graph is a colouring in which every two vertices sharing a common neighbour receive a different colour. Chen, Hahn, Raspaud and Wang conjectured that every planar graph of maximum degree $Δ\ge 3$ admits an injective colouring with at most $\lfloor 3Δ/2\rfloor$ colours. This was later disproved by Lužar and Škrekovski for certain small and even values of $Δ$ and they proposed a new refined conjecture. Using an algorithm for determining the injective chromatic number of a graph, i.e. the smallest number of colours for which the graph admits an injective colouring, we give computational evidence for Lužar and Škrekovski's conjecture and extend their results by presenting an infinite family of $3$-connected planar graphs for each $Δ$ (except for $4$) attaining their bound, whereas they only gave a finite amount of examples for each $Δ$. Hence, together with another infinite family of maximum degree $4$, we provide infinitely many counterexamples to the conjecture by Chen et al. for each $Δ$ if $4\le Δ\le 7$ and every even $Δ\ge 8$. We provide similar evidence for analogous conjectures by La and Štorgel and Lužar, Škrekovski and Tancer when the girth is restricted as well. Also in these cases we provide infinite families of $3$-connected planar graphs attaining the bounds of these conjectures for certain maximum degrees $Δ\geq 3$.

Figures (9)

• Figure 1: Planar graph of order $10$ with maximum degree $5$, diameter $2$ and injective chromatic number $10$.
• Figure 2: Planar graph with maximum degree $4$ and injective chromatic number $9$.
• Figure 3: Graph $G^4_0$ with $\Delta = 4$ and $\chi_i(G_0) = 8$.
• Figure 4: Visualisation of how to construct $G^{'\Delta}_i$ and $G^\Delta_i$ from $G^\Delta_{i-1}$ for a given triangle $v_0^{i-1}v_1^{i-1}v_2^{i-1}$ and $i\geq 1, \Delta\geq 5$.
• Figure 5: Base graphs for the infinite families constructed in Theorem \ref{['thm:inf_families']}, satisfying the requirements of Lemma \ref{['lemma:inf_family_procedure']} for the triangle $uvw$. Their maximum degree is $5$, $6$ and $7$, respectively. Note that the left graph is the same graph as Figure \ref{['fig:LS_correction']}.

Theorems & Definitions (29)

• Conjecture 1.1: Chen, Hahn, Raspaud, and Wang chen2012some
• Conjecture 1.2: Lužar and Škrekovski luvzar2015counterexamples
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Theorem 3.1
• proof
• Lemma 3.2
• proof