Injective edge-coloring of subcubic graphs

An injective edge-coloring [Formula: see text] of a graph [Formula: see text] is an edge-coloring such that if [Formula: see text], [Formula: see text], and [Formula: see text] are three consecutive edges in [Formula: see text] (they are consecutive if they form a path or a cycle of length three), then [Formula: see text] and [Formula: see text] receive different colors. The minimum integer [Formula: see text] such that, [Formula: see text] has an injective edge-coloring with [Formula: see text] colors, is called the injective chromatic index of [Formula: see text] ([Formula: see text]). This parameter was introduced by Cardoso et al. [Injective coloring of graphs, Filomat 33(19) (2019) 6411–6423, arXiv:1510.02626] motivated by the Packet Radio Network problem. They proved that computing [Formula: see text] of a graph [Formula: see text] is NP-hard. We give new upper bounds for this parameter and we present the relationships of the injective edge-coloring with other colorings of graphs. We study the injective edge-coloring of some classes of subcubic graphs. We prove that a subcubic bipartite graph has an injective chromatic index bounded by [Formula: see text]. We also prove that if [Formula: see text] is a subcubic graph with maximum average degree less than [Formula: see text] (respectively, [Formula: see text]), then [Formula: see text] admits an injective edge-coloring with at most 4 (respectively, [Formula: see text]) colors. Moreover, we establish a tight upper bound for subcubic outerplanar graphs.

Homomorphisms of Sparse Signed Graphs

The notion of homomorphism of signed graphs, introduced quite recently, provides better interplay with the notion of minor and is thus of high importance in graph coloring. A newer, but equivalent, definition of homomorphisms of signed graphs, proposed jointly by the second and third authors of this paper and Thomas Zaslavsky, leads to a basic no-homomorphism lemma. According to this definition, a signed graph $(G, \sigma)$ admits a homomorphism to a signed graph $(H, \pi)$ if there is a mapping $\phi$ from the vertices and edges of $G$ to the vertices and edges of $H$ (respectively) which preserves adjacencies, incidences, and signs of closed walks (i.e., the product of the sign of their edges). For $ij=00, 01, 10, 11$, let $g_{ij}(G,\sigma)$ be the length of a shortest nontrivial closed walk of $(G, \sigma)$ which is, positive and of even length for $ij=00$, positive and of odd length for $ij=01$, negative and of even length for $ij=10$, negative and of odd length for $ij=11$. For each $ij$, if there is no nontrivial closed walk of the corresponding type, we let $g_{ij}(G, \sigma)=\infty$. If $G$ is bipartite, then $g_{01}(G,\sigma)=g_{11}(G,\sigma)=\infty$. In this case, $g_{10}(G,\sigma)$ is certainly realized by a cycle of $G$, and it will be referred to as the \emph{unbalanced-girth} of $(G,\sigma)$. It then follows that if $(G,\sigma)$ admits a homomorphism to $(H, \pi)$, then $g_{ij}(G, \sigma)\geq g_{ij}(H, \pi)$ for $ij \in \{00, 01,10,11\}$. Studying the restriction of homomorphisms of signed graphs on sparse families, in this paper we first prove that for any given signed graph $(H, \pi)$, there exists a positive value of $\epsilon$ such that, if $G$ is a connected graph of maximum average degree less than $2+\epsilon$, and if $\sigma$ is a signature of $G$ such that $g_{ij}(G, \sigma)\geq g_{ij}(H, \pi)$ for all $ij \in \{00, 01,10,11\}$, then $(G, \sigma)$ admits a homomorphism to $(H, \pi)$. For $(H, \pi)$ being the signed graph on $K_4$ with exactly one negative edge, we show that $\epsilon=\frac{4}{7}$ works and that this is the best possible value of $\epsilon$. For $(H, \pi)$ being the negative cycle of length $2g$, denoted $UC_{2g}$, we show that $\epsilon=\frac{1}{2g-1}$ works. As a bipartite analogue of the Jaeger-Zhang conjecture, Naserasr, Sopena and Rollovà conjectured in [Homomorphisms of signed graphs, {\em J. Graph Theory} 79 (2015)] that every signed bipartite planar graph $(G,\sigma)$ satisfying $g_{ij}(G,\sigma)\geq 4g-2$ admits a homomorphism to $UC_{2g}$. We show that $4g-2$ cannot be strengthened, and, supporting the conjecture, we prove it for planar signed bipartite graphs $(G,\sigma)$ satisfying the weaker condition $g_{ij}(G,\sigma)\geq 8g-2$. In the course of our work, we also provide a duality theorem to decide whether a 2-edge-colored graph admits a homomorphism to a certain class of 2-edge-colored signed graphs or not.

Geometric recognition methodology of honeycomb structure based on dynamic window

Defect determination is important for guaranteeing the quality of sandwich structure. The present study, a geometric recognition methodology based on dynamic window was constructed for honeycomb structure, by identifying the vertices from given images captured in the manufacturing process. The combination of traversing window, annihilation window, and tracking window was developed to reconstruct the contoured maximums, locate the vertices positions, and determine the cell–node relationships, respectively. Moore boundary tracking was practically used to deal with the problem caused by the edge effect. Based on these, all relevant vertices as well as their relationship of each honeycomb cell can be determined. Afterward, quality assessment of the honeycomb product was carried out in terms of the maximum, the average, and the maximum average degree deviation. Illustrations validate the present methodology well. All these achievements shed a light on design of the consistency, reliability, and homogeneity of high-standard sandwich structure.

Defective and clustered choosability of sparse graphs

An (improper) graph colouring hasdefect dif each monochromatic subgraph has maximum degree at mostd, and hasclustering cif each monochromatic component has at mostcvertices. This paper studies defective and clustered list-colourings for graphs with given maximum average degree. We prove that every graph with maximum average degree less than (2d+2)/(d+2)kisk-choosable with defectd. This improves upon a similar result by Havet and Sereni (J. Graph Theory, 2006). For clustered choosability of graphs with maximum average degreem, no (1-ɛ)mbound on the number of colours was previously known. The above result withd=1 solves this problem. It implies that every graph with maximum average degreemis$\lfloor{\frac{3}{4}m+1}\rfloor$-choosable with clustering 2. This extends a result of Kopreski and Yu (Discrete Math., 2017) to the setting of choosability. We then prove two results about clustered choosability that explore the trade-off between the number of colours and the clustering. In particular, we prove that every graph with maximum average degreemis$\lfloor{\frac{7}{10}m+1}\rfloor$-choosable with clustering 9, and is$\lfloor{\frac{2}{3}m+1}\rfloor$-choosable with clusteringO(m). As an example, the later result implies that every biplanar graph is 8-choosable with bounded clustering. This is the best known result for the clustered version of the earth–moon problem. The results extend to the setting where we only consider the maximum average degree of subgraphs with at least some number of vertices. Several applications are presented.