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A star edge coloring of a graph G is a proper edge coloring with no 2-colored path or cycle of length four. The star edge coloring problem is to find an edge coloring of a given graph G with minimum number k of colors such that G admits a star ...

A k-proper connected 2-coloring for a graph is an edge bipartition which ensures the existence of at least k vertex disjoint simple alternating paths (i.e., paths where no two adjacent edges belong to the same partition) between all pairs of ...${\mathbb{}}_{\mathrm{}}$$$$$$\mathrm{}$${\mathbb{}}_{\mathrm{}}$${\mathrm{}}^{\left(\frac{}{{}^{\mathrm{}}}\right)}$${\mathbb{}}_{\mathrm{}}$$\mathrm{}$${\mathrm{}}^{\left(\frac{}{{}^{\mathrm{}}}\right)}{}^{\mathrm{}}$$\mathrm{}$

We provide new deterministic algorithms for the edge coloring problem, which is one of the classic and highly studied distributed local symmetry breaking problems. As our main result, we show that a (2Δ - 1)-edge coloring can be computed in time poly ...

The problem of coloring the edges of an n-node graph of maximum degree Δ with 2Δ − 1 colors is one of the key symmetry breaking problems in the area of distributed graph algorithms. While there has been a lot of progress towards the understanding of ...

In this paper, we first study a new extremal problem recently posed by Conlon and Tyomkyn [Repeated Patterns in Proper Colourings, preprint, https://arxiv.org/abs/2002.00921 (2020)]. Given a graph $H$ and an integer $k\geqslant 2$, let $f_{k}(n,H)$ be the ...

The complexity of distributed edge coloring depends heavily on the palette size as a function of the maximum degree Δ. In this article, we explore the complexity of edge coloring in the LOCAL model in different palette size regimes. Our results are as ...

Vizing showed that it suffices to color the edges of a simple graph using Δ + 1 colors, where Δ is the maximum degree of the graph. However, up to this date, no efficient distributed edge-coloring algorithm is known for obtaining such coloring, even for ...

The problem of inferring unknown parameters of a networked social system is of considerable practical importance. We consider this problem for the independent cascade model using an active query framework. More specifically, given a network whose edge ...

AbstractA proper edge coloring of a graph G with colors 1,2,⋯,t is called a cyclic interval t‐coloring if for each vertex v of G the edges incident to v are colored by consecutive colors, under the condition that color 1 is considered as consecutive to ...

For a graph $G$, let $\nu_s(G)$ be the strong matching number of $G$. We prove the sharp bound $\nu_s(G)\geq \frac{n(G)}{9}$ for every graph $G$ of maximum degree at most 4 and without isolated vertices that does not contain a certain blown-up 5-cycle as a ...

In the article, the existence of rainbow cycles in edge colored plane triangulations is studied. It is shown that the minimum number rb Tn,C3 of colors that force the existence of a rainbow C3 in any n-vertex plane triangulation is equal to ï 3n-42ï . ...

An edge colored graph $G$ is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order ...

We show that an effective version of Siegel's Theorem on finiteness of integer solutions for a specific algebraic curve and an application of elementary Galois theory are key ingredients in a complexity classification of some Holant problems. These ...

Drawings of non-planar graphs always result in edge crossings.When there are many edges crossing at small angles, it is often difficult to follow these edges, because of the multiple visual paths resulted from the crossings that slow down eye movements. ...

Let $G=(V,E)$ be a simple graph of maximum degree $\Delta$. The edges of $G$ can be colored with at most $\Delta +1$ colors by Vizing's theorem. We study lower bounds on the size of subgraphs of $G$ that can be colored with $\Delta$ colors. Vizing's ...

AbstractIt was conjectured by Fan and Raspaud 1994 that every bridgeless cubic graph contains three perfect matchings such that every edge belongs to at most two of them. We show a randomized algorithmic way of finding Fan-Raspaud colorings of a given ...

The general Bandpass-<em>B</em> problem is NP-hard and can be approximated by a reduction into the <em>B</em> -set packing problem, with a worst case performance ratio of <em>O</em> (<em>B</em> ²). When <em>B</em> =2, a maximum weight matching gives a 2-...

In 1968, Vizing conjectured that if G is a Δ-critical graph with n vertices, then α(G)≤n/2, where α(G) is the independence number of G. In this paper, we apply Vizing and Vizing-like adjacency lemmas to this problem and prove that α(G)<(((5Δ−6)n)/(8Δ−6))...

A graph G with maximum degree Δ and edge chromatic number χ′(G)>Δ is edge-Δ-critical if χ′(G−e)=Δ for every edge e of G. It is proved here that the vertex independence number of an edge-Δ-critical graph of order n is less than **image**. For large Δ, ...

A sequence r₁, r₂, …, r_2n such that r_i=r_{n+ i} for all 1≤i≤n is called a repetition. A sequence S is called non-repetitive if no block (i.e. subsequence of consecutive terms of S) is a repetition. Let G be a graph whose edges are colored. A trail is ...