Given a graph G and a mapping f:V(G)→N, an f-list assignment of G is a function that maps each v∈V(G) to a set of at least f(v) colors. For an f-list assignment L of a graph G, a proper conflict-free L-coloring of G is a proper coloring ϕ of G such that for every non-isolated vertex v∈V(G), ϕ(v)∈L(v) and some color appears precisely once in the neighborhood of v. We say that G is proper conflict-free f-choosable if for every f-list assignment L of G, there exists a proper conflict-free L-coloring of G. If G is proper conflict-free f-choosable and there is a constant k such that f(v)=dG(v)+k for every vertex v of G, then we say G is proper conflict-free (degree+k)-choosable. In this paper, we consider graphs with a bounded maximum average degree. We show that every graph with the maximum average degree less than 103 is proper conflict-free (degree+3)-choosable, and that every graph with the maximum average degree less than 187 is proper conflict-free (degree+2)-choosable unless it contains a component isomorphic to the 5-cycle. As a result, every planar graph with girth at least 5 is proper conflict-free (degree+3)-choosable, and every planar graph with girth at least 9 is proper conflict-free (degree+2)-choosable.
Given a family of graphs ℱ, a graph G is ℱ-saturated if it is ℱ-free but the addition of any missing edge creates a copy of some F ∈ℱ. The study of the minimum number of edges in ℱ-saturated graphs is a central topic in extremal graph theory. Let (p+1)K_2 denote a matching of size p+1. Determining the minimum number of edges in a (p+1)K_2-saturated graph is a fundamental question in this area, explicitly posed as Problem 9 in the survey by Faudree et al. (2011). In this paper, we refine the structural analysis of (p+1)K_2-saturated graphs and derive an explicit formula for the number of edges in terms of a single integer parameter. By minimizing this formula we determine sat(n,(p+1)K_2) for all n>2p, thereby resolving Problem 9 in full generality and extending earlier results of Kászonyi–Tuza (1986) and Zhang–Lu–Yu (2024). Moreover, by maximizing the same formula we recover the classical Erdős–Gallai (1959) upper bound on the number of edges in such graphs.
An r-regular graph is an r-graph, if every odd set of vertices is connected to its complement by at least r edges. We prove for r ∈{4,5}, every projective planar r-graph with no Petersen-minor is r-edge colorable.
Existing meta-path-based recommendation systems suffer from noise interference and intent fusion, which distort high-order semantic relationships and consequently degrade recommendation performance. To tackle these challenges, we propose meta-path Diffusion Denoising and Multi-view Contrastive Learning for recommendation (DDMCL), which enhances representation learning from three perspectives: (1) Dual-graph diffusion denoising: we employ diffusion model to simultaneously reconstruct the latent semantic distributions of two meta-path graphs on the user (item) side, achieving joint optimization through a shared-parameter network with graph-specific identifiers to enhance denoising performance while preserving semantic consistency. (2) Intent attention fusion: we extract path semantic representations based on graph convolutional networks, design an attention mechanism to dynamically aggregate intent features, and use the embedding of the user-item graph as a structural guide to capture the multi-dimensional preferences of users and items. (3) Multi-view contrastive optimization: we perform intent-aware contrastive learning in the meta-path semantic space and structure-enhanced contrastive learning in the user-item interaction space, jointly optimized with the main recommendation objective to improve robustness and generalization. Extensive experiments on six benchmark recommendation datasets demonstrate that DDMCL outperforms state-of-the-art baselines, improving the NDCG@20 on the Amazon and Yelp datasets by 3.27% and 3.70%, respectively.
