Vizing's Theorem and Edge-Chromatic Graph
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A Brief History of Edge-Colorings — with Personal Reminiscences
Discrete Mathematics Letters Discrete Math. Lett. 6 (2021) 38–46 www.dmlett.com DOI: 10.47443/dml.2021.s105 Review Article A brief history of edge-colorings – with personal reminiscences∗ Bjarne Toft1;y, Robin Wilson2;3 1Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark 2Department of Mathematics and Statistics, Open University, Walton Hall, Milton Keynes, UK 3Department of Mathematics, London School of Economics and Political Science, London, UK (Received: 9 June 2020. Accepted: 27 June 2020. Published online: 11 March 2021.) c 2021 the authors. This is an open access article under the CC BY (International 4.0) license (www.creativecommons.org/licenses/by/4.0/). Abstract In this article we survey some important milestones in the history of edge-colorings of graphs, from the earliest contributions of Peter Guthrie Tait and Denes´ Konig¨ to very recent work. Keywords: edge-coloring; graph theory history; Frank Harary. 2020 Mathematics Subject Classification: 01A60, 05-03, 05C15. 1. Introduction We begin with some basic remarks. If G is a graph, then its chromatic index or edge-chromatic number χ0(G) is the smallest number of colors needed to color its edges so that adjacent edges (those with a vertex in common) are colored differently; for 0 0 0 example, if G is an even cycle then χ (G) = 2, and if G is an odd cycle then χ (G) = 3. For complete graphs, χ (Kn) = n−1 if 0 0 n is even and χ (Kn) = n if n is odd, and for complete bipartite graphs, χ (Kr;s) = max(r; s). -
When the Vertex Coloring of a Graph Is an Edge Coloring of Its Line Graph — a Rare Coincidence
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repository of the Academy's Library When the vertex coloring of a graph is an edge coloring of its line graph | a rare coincidence Csilla Bujt¶as 1;¤ E. Sampathkumar 2 Zsolt Tuza 1;3 Charles Dominic 2 L. Pushpalatha 4 1 Department of Computer Science and Systems Technology, University of Pannonia, Veszpr¶em,Hungary 2 Department of Mathematics, University of Mysore, Mysore, India 3 Alfr¶edR¶enyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary 4 Department of Mathematics, Yuvaraja's College, Mysore, India Abstract The 3-consecutive vertex coloring number Ã3c(G) of a graph G is the maximum number of colors permitted in a coloring of the vertices of G such that the middle vertex of any path P3 ½ G has the same color as one of the ends of that P3. This coloring constraint exactly means that no P3 subgraph of G is properly colored in the classical sense. 0 The 3-consecutive edge coloring number Ã3c(G) is the maximum number of colors permitted in a coloring of the edges of G such that the middle edge of any sequence of three edges (in a path P4 or cycle C3) has the same color as one of the other two edges. For graphs G of minimum degree at least 2, denoting by L(G) the line graph of G, we prove that there is a bijection between the 3-consecutive vertex colorings of G and the 3-consecutive edge col- orings of L(G), which keeps the number of colors unchanged, too. -
Strongly Connected Component
Graph IV Ian Things that we would talk about ● DFS ● Tree ● Connectivity Useful website http://codeforces.com/blog/entry/16221 Recommended Practice Sites ● HKOJ ● Codeforces ● Topcoder ● Csacademy ● Atcoder ● USACO ● COCI Term in Directed Tree ● Consider node 4 – Node 2 is its parent – Node 1, 2 is its ancestors – Node 5 is its sibling – Node 6 is its child – Node 6, 7, 8 is its descendants ● Node 1 is the root DFS Forest ● When we do DFS on a graph, we would obtain a DFS forest. Noted that the graph is not necessarily a tree. ● Some of the information we get through the DFS is actually very useful, such as – Starting time of a node – Finishing time of a node – Parent of the node Some Tricks Using DFS Order ● Suppose vertex v is ancestor(not only parent) of u – Starting time of v < starting time of u – Finishing time of v > starting time of u ● st[v] < st[u] <= ft[u] < ft[v] ● O(1) to check if ancestor or not ● Flatten the tree to store subtree information(maybe using segment tree or other data structure to maintain) ● Super useful !!!!!!!!!! Partial Sum on Tree ● Given queries, each time increase all node from node v to node u by 1 ● Assume node v is ancestor of node u ● sum[u]++, sum[par[v]]-- ● Run dfs in root dfs(v) for all child u dfs(u) d[v] = d[v] + d[u] Types of Edges ● Tree edges – Edges that forms a tree ● Forward edges – Edges that go from a node to its descendants but itself is not a tree edge. -
Interval Edge-Colorings of Graphs
University of Central Florida STARS Electronic Theses and Dissertations, 2004-2019 2016 Interval Edge-Colorings of Graphs Austin Foster University of Central Florida Part of the Mathematics Commons Find similar works at: https://stars.library.ucf.edu/etd University of Central Florida Libraries http://library.ucf.edu This Masters Thesis (Open Access) is brought to you for free and open access by STARS. It has been accepted for inclusion in Electronic Theses and Dissertations, 2004-2019 by an authorized administrator of STARS. For more information, please contact [email protected]. STARS Citation Foster, Austin, "Interval Edge-Colorings of Graphs" (2016). Electronic Theses and Dissertations, 2004-2019. 5133. https://stars.library.ucf.edu/etd/5133 INTERVAL EDGE-COLORINGS OF GRAPHS by AUSTIN JAMES FOSTER B.S. University of Central Florida, 2015 A thesis submitted in partial fulfilment of the requirements for the degree of Master of Science in the Department of Mathematics in the College of Sciences at the University of Central Florida Orlando, Florida Summer Term 2016 Major Professor: Zixia Song ABSTRACT A proper edge-coloring of a graph G by positive integers is called an interval edge-coloring if the colors assigned to the edges incident to any vertex in G are consecutive (i.e., those colors form an interval of integers). The notion of interval edge-colorings was first introduced by Asratian and Kamalian in 1987, motivated by the problem of finding compact school timetables. In 1992, Hansen described another scenario using interval edge-colorings to schedule parent-teacher con- ferences so that every person’s conferences occur in consecutive slots. -
CLRS B.4 Graph Theory Definitions Unit 1: DFS Informally, a Graph
CLRS B.4 Graph Theory Definitions Unit 1: DFS informally, a graph consists of “vertices” joined together by “edges,” e.g.,: example graph G0: 1 ···················•······························· ····························· ····························· ························· ···· ···· ························· ························· ···· ···· ························· ························· ···· ···· ························· ························· ···· ···· ························· ············· ···· ···· ·············· 2•···· ···· ···· ··· •· 3 ···· ···· ···· ···· ···· ··· ···· ···· ···· ···· ···· ···· ···· ···· ···· ···· ······· ······· ······· ······· ···· ···· ···· ··· ···· ···· ···· ···· ···· ··· ···· ···· ···· ···· ···· ···· ···· ···· ···· ···· ··············· ···· ···· ··············· 4•························· ···· ···· ························· • 5 ························· ···· ···· ························· ························· ···· ···· ························· ························· ···· ···· ························· ····························· ····························· ···················•································ 6 formally a graph is a pair (V, E) where V is a finite set of elements, called vertices E is a finite set of pairs of vertices, called edges if H is a graph, we can denote its vertex & edge sets as V (H) & E(H) respectively if the pairs of E are unordered, the graph is undirected if the pairs of E are ordered the graph is directed, or a digraph two vertices joined by an edge -
Graph Theory
1 Graph Theory “Begin at the beginning,” the King said, gravely, “and go on till you come to the end; then stop.” — Lewis Carroll, Alice in Wonderland The Pregolya River passes through a city once known as K¨onigsberg. In the 1700s seven bridges were situated across this river in a manner similar to what you see in Figure 1.1. The city’s residents enjoyed strolling on these bridges, but, as hard as they tried, no residentof the city was ever able to walk a route that crossed each of these bridges exactly once. The Swiss mathematician Leonhard Euler learned of this frustrating phenomenon, and in 1736 he wrote an article [98] about it. His work on the “K¨onigsberg Bridge Problem” is considered by many to be the beginning of the field of graph theory. FIGURE 1.1. The bridges in K¨onigsberg. J.M. Harris et al., Combinatorics and Graph Theory , DOI: 10.1007/978-0-387-79711-3 1, °c Springer Science+Business Media, LLC 2008 2 1. Graph Theory At first, the usefulness of Euler’s ideas and of “graph theory” itself was found only in solving puzzles and in analyzing games and other recreations. In the mid 1800s, however, people began to realize that graphs could be used to model many things that were of interest in society. For instance, the “Four Color Map Conjec- ture,” introduced by DeMorgan in 1852, was a famous problem that was seem- ingly unrelated to graph theory. The conjecture stated that four is the maximum number of colors required to color any map where bordering regions are colored differently. -
Superposition and Constructions of Graphs Without Nowhere-Zero K-flows
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Europ. J. Combinatorics (2002) 23, 281–306 doi:10.1006/eujc.2001.0563 Available online at http://www.idealibrary.com on Superposition and Constructions of Graphs Without Nowhere-zero k-flows M ARTIN KOCHOL Using multi-terminal networks we build methods on constructing graphs without nowhere-zero group- and integer-valued flows. In this way we unify known constructions of snarks (nontrivial cubic graphs without edge-3-colorings, or equivalently, without nowhere-zero 4-flows) and provide new ones in the same process. Our methods also imply new complexity results about nowhere-zero flows in graphs and state equivalences of Tutte’s 3- and 5-flow conjectures with formally weaker statements. c 2002 Elsevier Science Ltd. All rights reserved. 1. INTRODUCTION Nowhere-zero flows in graphs have been introduced by Tutte [38–40]. Primarily he showed that a planar graph is face-k-colorable if and only if it admits a nowhere-zero k-flow (its edges can be oriented and assigned values ±1,..., ±(k − 1) so that the sum of the incoming values equals the sum of the outcoming ones for every vertex of the graph). Tutte also proved the classical equivalence result that a graph admits a nowhere-zero k-flow if and only if it admits a flow whose values are the nonzero elements of a finite abelian group of order k. Seymour [35] has proved that every bridgeless graph admits a nowhere-zero 6-flow, thereby improving the 8-flow theorem of Jaeger [16] and Kilpatrick [20]. -
Finding Articulation Points and Bridges Articulation Points Articulation Point
Finding Articulation Points and Bridges Articulation Points Articulation Point Articulation Point A vertex v is an articulation point (also called cut vertex) if removing v increases the number of connected components. A graph with two articulation points. 3 / 1 Articulation Points Given I An undirected, connected graph G = (V; E) I A DFS-tree T with the root r Lemma A DFS on an undirected graph does not produce any cross edges. Conclusion I If a descendant u of a vertex v is adjacent to a vertex w, then w is a descendant or ancestor of v. 4 / 1 Removing a Vertex v Assume, we remove a vertex v 6= r from the graph. Case 1: v is an articulation point. I There is a descendant u of v which is no longer reachable from r. I Thus, there is no edge from the tree containing u to the tree containing r. Case 2: v is not an articulation point. I All descendants of v are still reachable from r. I Thus, for each descendant u, there is an edge connecting the tree containing u with the tree containing r. 5 / 1 Removing a Vertex v Problem I v might have multiple subtrees, some adjacent to ancestors of v, and some not adjacent. Observation I A subtree is not split further (we only remove v). Theorem A vertex v is articulation point if and only if v has a child u such that neither u nor any of u's descendants are adjacent to an ancestor of v. Question I How do we determine this efficiently for all vertices? 6 / 1 Detecting Descendant-Ancestor Adjacency Lowpoint The lowpoint low(v) of a vertex v is the lowest depth of a vertex which is adjacent to v or a descendant of v. -
Acyclic Edge-Coloring of Planar Graphs
Acyclic edge-coloring of planar graphs: ∆ colors suffice when ∆ is large Daniel W. Cranston∗ January 31, 2019 Abstract An acyclic edge-coloring of a graph G is a proper edge-coloring of G such that the subgraph ′ induced by any two color classes is acyclic. The acyclic chromatic index, χa(G), is the smallest ′ number of colors allowing an acyclic edge-coloring of G. Clearly χa(G) ∆(G) for every graph G. Cohen, Havet, and M¨uller conjectured that there exists a constant M≥such that every planar ′ graph with ∆(G) M has χa(G) = ∆(G). We prove this conjecture. ≥ 1 Introduction proper A proper edge-coloring of a graph G assigns colors to the edges of G such that two edges receive edge- distinct colors whenever they have an endpoint in common. An acyclic edge-coloring is a proper coloring acyclic edge-coloring such that the subgraph induced by any two color classes is acyclic (equivalently, the edge- ′ edges of each cycle receive at least three distinct colors). The acyclic chromatic index, χa(G), is coloring the smallest number of colors allowing an acyclic edge-coloring of G. In an edge-coloring ϕ, if a acyclic chro- color α is used incident to a vertex v, then α is seen by v. For the maximum degree of G, we write matic ∆(G), and simply ∆ when the context is clear. Note that χ′ (G) ∆(G) for every graph G. When index a ≥ we write graph, we forbid loops and multiple edges. A planar graph is one that can be drawn in seen by planar the plane with no edges crossing. -
NP-Completeness of List Coloring and Precoloring Extension on the Edges of Planar Graphs
NP-completeness of list coloring and precoloring extension on the edges of planar graphs D´aniel Marx∗ 17th October 2004 Abstract In the edge precoloring extension problem we are given a graph with some of the edges having a preassigned color and it has to be decided whether this coloring can be extended to a proper k-edge-coloring of the graph. In list edge coloring every edge has a list of admissible colors, and the question is whether there is a proper edge coloring where every edge receives a color from its list. We show that both problems are NP-complete on (a) planar 3-regular bipartite graphs, (b) bipartite outerplanar graphs, and (c) bipartite series-parallel graphs. This improves previous results of Easton and Parker [6], and Fiala [8]. 1 Introduction In graph vertex coloring we have to assign colors to the vertices such that neighboring vertices receive different colors. Starting with [7] and [28], a gener- alization of coloring was investigated: in the list coloring problem each vertex can receive a color only from its prescribed list of admissible colors. In the pre- coloring extension problem a subset W of the vertices have preassigned colors and we have to extend this precoloring to a proper coloring of the whole graph, using only colors from a given color set C. It can be viewed as a special case of list coloring: the list of a precolored vertex consists of a single color, while the list of every other vertex is C. A thorough survey on list coloring, precoloring extension, and list chromatic number can be found in [26, 1, 12, 13]. -
31 Aug 2020 Injective Edge-Coloring of Sparse Graphs
Injective edge-coloring of sparse graphs Baya Ferdjallah1,4, Samia Kerdjoudj 1,5, and André Raspaud2 1LIFORCE, Faculty of Mathematics, USTHB, BP 32 El-Alia, Bab-Ezzouar 16111, Algiers, Algeria 2LaBRI, Université de Bordeaux, 351 cours de la Libération, 33405 Talence Cedex, France 4Université de Boumerdès, Avenue de l’indépendance, 35000, Boumerdès, Algeria 5Université de Blida 1, Route de Soumâa BP 270, Blida, Algeria September 1, 2020 Abstract An injective edge-coloring c of a graph G is an edge-coloring such that if e1, e2, and e3 are three consecutive edges in G (they are consecutive if they form a path or a cycle of length three), then e1 and e3 receive different colors. The minimum integer k such that, G has an injective edge-coloring with k ′ colors, is called the injective chromatic index of G (χinj(G)). This parameter was introduced by Cardoso et al. [12] motivated by the Packet Radio Network ′ problem. They proved that computing χinj(G) of a graph G is NP-hard. We give new upper bounds for this parameter and we present the rela- tionships of the injective edge-coloring with other colorings of graphs. The obtained general bound gives 8 for the injective chromatic index of a subcubic graph. If the graph is subcubic bipartite we improve this last bound. We prove that a subcubic bipartite graph has an injective chromatic index bounded by arXiv:1907.09838v2 [math.CO] 31 Aug 2020 6. We also prove that if G is a subcubic graph with maximum average degree 7 8 less than 3 (resp. -
A Multigraph May Contain Parallel Edges. Definition 1 Edge Coloring Of
Edge Coloring A multigraph may contain parallel edges. Definition 1 Edge coloring of a graph G is an assignment of colors to its edges so that adjacent edges are assigned distinct colors. Definition 2 The chromatic index, or the edge-chromatic number of a graph G is the smallest integer n for which there is an edge coloring of G which uses n colors. 1 Edge coloring is a special case of vertex coloring. Definition 3 For a given G = (V,E), the line graph L(G) is a graph H whose set of vertices is E and the set of edges consists of all pairs (ei,e2) where e1 and e2 are adjacent edges in G. Theorem 1 For a simple graph H, there is a solution G to L(G)= H iff H decomposes into complete subgraphs, with each vertex of H appearing in at most two of these complete subgraphs. Proposition 1 Every edge coloring is a partition of the edges into matchings. Proposition 2 There is an edge-coloring of G in n colors iff there is a vertex coloring of L(G) which uses n colors. 2 For a graph G, χ′(G) and ∆(G) denote the chromatic index and the maximum vertex degree respectively. Theorem 2 ∀G, χ′(G) ≥ ∆(G). Proof. Obvious. Theorem 3 ∀G, χ′(G) ≤ 2∆(G) − 1. Proof. Apply the greedy algorithm: color edges one-by-one, using for each edge the smallest positive integer which is available. Theorem 4 ∀G, if G is bipartite, then χ′(G)=∆(G). Proof. Not obvious.