Star colouring and locally constrained graph homomorphisms

TL;DR

This work advances the understanding of star colourings in even-degree regular graphs by embedding them in a framework of locally constrained graph homomorphisms to oriented line graphs. It introduces LBH and OBH as driving concepts and leverages the oriented line graph operation $\vec{L}(\cdot)$ and its undirected counterpart $L^*(\cdot)$ to connect colouring, orientation, and lifting properties. For $p\ge 2$, it proves that a $K_{1,p+1}$-free $2p$-regular graph is $(p+2)$-star colourable if and only if it admits a locally bijective homomorphism to $L^*(K_{p+2})$ and equivalently an orientation with an OBH to $\vec{L}(K_{p+2})$, yielding eigenvalue consequences and providing both structural and complexity results (e.g., NP-completeness in planar cases and a characterization for line graphs of 3-regular graphs). The paper also outlines several open problems and future directions, including broader classifications of $(2p-1)$-regular cases and the spectrum of $L^*(G)$, highlighting the practical impact on graph colouring, spectral theory, and algorithmic complexity within graph theory.

Abstract

We relate star colouring of even-degree regular graphs to the notions of locally constrained graph homomorphisms to the oriented line graph $ \vec{L}(K_q) $ of the complete graph $ K_q $ and to its underlying undirected graph $ L^*(K_q) $. Our results have consequences for locally constrained graph homomorphisms and oriented line graphs in addition to star colouring. We show that $ L^*(H) $ is a 2-lift of the line graph $ L(H) $ for every graph $ H $. Dvořák, Mohar and Šámal (J. Graph Theory, 2013) proved that for every 3-regular graph $ G $, the line graph of $ G $ is 4-star colourable if and only if $ G $ admits a locally bijective homomorphism to the cube $ Q_3 $. We generalise this result as follows: for $ p\geq 2 $, a $ K_{1,p+1} $-free $ 2p $-regular graph $ G $ admits a $ (p+2) $-star colouring if and only if $ G $ admits a locally bijective homomorphism to $ L^*(K_{p+2}) $. As a result, if a $ K_{p+1} $-free $ 2p $-regular graph $ G $ with $ p\geq 2 $ is $ (p+2) $-star colourable, then $ -2 $ and $ p-2 $ are eigenvalues of $ G $. We also prove the following: (i) for $ p\geq 2 $, a $ 2p $-regular graph $ G $ admits a $ (p+2) $-star colouring if and only if $ G $ has an orientation that admits an out-neighbourhood bijective homomorphism to $ \vec{L}(K_{p+2}) $; (ii) the line graph of a 3-regular graph $ G $ is 4-star colourable if and only if $ G $ is bipartite and distance-two 4-colourable; and (iii) it is NP-complete to check whether a planar 4-regular 3-connected graph is 4-star colourable.

Figures (7)

• Figure 1: A locally bijective homomorphism from a graph $G$ to a graph $H$. The vertices in $H$ are labelled distinct and are drawn by distinct shapes. For each vertex $w$ of $H$, each copy of $w$ in $G$ is drawn in the same shape as $w$ (and labelled the same).
• Figure 2: An out-neighbourhood bijective homomorphism from an oriented graph $\vec{G}$ to an oriented graph $\vec{H}$. The vertices in $\vec{H}$ are labelled distinct and are drawn by distinct shapes. For each vertex $w$ of $\vec{H}$, each copy of $w$ in $\vec{G}$ is drawn in the same shape as $w$ (and labelled the same).
• Figure 3: A 4-coloured MINI-orientation of $L(Q_3)$ (the cuboctahedral graph).
• Figure 4: An example of the oriented line graph operation.
• Figure 5: Possible orientations of a $C_4$ in a coloured MINI-orientation of $G$.

Theorems & Definitions (61)

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