《图论》课程教学资源（书籍文献）Color-Induced Graph Colorings

SPRINGER BRIEFS IN MATHEMATICS Ping Zhang Color-Induced Graph Colorings ②Springer

123 SPRINGER BRIEFS IN MATHEMATICS Ping Zhang Color-Induced Graph Colorings

Preface The interest inedge colorings of graphs can be traced back to 1880 when the Scottish mathematician Peter Guthrie Tait attempted to solve the Four Color Problem with the aid of edge colorings.Despite the fact that Tait's approach was not successful. it initiated a new concept.In 1964,Vadim Vizing proved that the minimum number of colors needed to color the edges of a graph so that every two adjacent edges are ly (popedec)one of This 之二onl In recent decades,there has been great interest in edge colorings that give rise to vertex colorings in a variety of ways,which is the subject of this book.While we will be describing many ways in which edge colorings have induced vertex colorings he major results.problems,and conjectures that have arisen in our goal to give a detailed survey of these subjects.Indeed.it is our intention to of s nt colo avenues of research topics In Chap.1,we begin with a brief review of the well-known concepts of proper edge colorings and proper vertex colorings,including many fundamental results concerning them. In Chap.2,unrestricted edge colorings of graphs are considered whose colors the setNof positive int gers or a set[=1.2.....k for some such an edge colo h G a sum-defined ver is defined olor()ofv is the st of the edges inciden The dge coloring cis or irregular if the resulting vertex coloring chas the property that c() every pair u.v of distinct vertices of G.The minimum positive integer k for which a graph G has such a vertex-distinguishing edge coloring is the irregularity strength of G.In Chap.3,the corresponding coloring is considered in which the colors are taken from a set Z of integers modulo k

Preface The interest in edge colorings of graphs can be traced back to 1880 when the Scottish mathematician Peter Guthrie Tait attempted to solve the Four Color Problem with the aid of edge colorings. Despite the fact that Tait’s approach was not successful, it initiated a new concept. In 1964, Vadim Vizing proved that the minimum number of colors needed to color the edges of a graph so that every two adjacent edges are colored differently (proper edge colorings) is one of two numbers, namely either the maximum degree or the maximum degree plus one. This result led to an increased interest and study of edge colorings in graph theory, not only edge colorings that are proper but also edge colorings that are not. In recent decades, there has been great interest in edge colorings that give rise to vertex colorings in a variety of ways, which is the subject of this book. While we will be describing many ways in which edge colorings have induced vertex colorings and some of the major results, problems, and conjectures that have arisen in this area of study, it is not our goal to give a detailed survey of these subjects. Indeed, it is our intention to provide an organized summary of several recent coloring concepts and topics that belong to this area of study, with the hope that this may suggest new avenues of research topics. In Chap. 1, we begin with a brief review of the well-known concepts of proper edge colorings and proper vertex colorings, including many fundamental results concerning them. In Chap. 2, unrestricted edge colorings of graphs are considered whose colors are elements of the set N of positive integers or a set Œk D f1; 2; : : : ; kg for some positive integer k. From such an edge coloring c of a graph G, a sum-defined vertex coloring c0 is defined, that is, for each vertex v of G, the color c0 .v/ of v is the sum of the colors of the edges incident with v. The edge coloring c is vertex-distinguishing or irregular if the resulting vertex coloring c0 has the property that c0 .u/ ¤ c0 .v/ for every pair u; v of distinct vertices of G. The minimum positive integer k for which a graph G has such a vertex-distinguishing edge coloring is the irregularity strength of G. In Chap. 3, the corresponding coloring is considered in which the colors are taken from a set Zk of integers modulo k. v

vi Preface Chapter 4 also deals with unrestricted edge coloringsc of graphs but here the induced vertex coloring is defined so that the color c(v)of a vertex v is the set of colors of its incident edges.In Chap.5,the emphasis changes from vertex colorings that are set-defined to those that are multiset-defined.In both cases.the induced vertex colorings c'are vertex-distinguishing. In Chap.6.unrestricted edge colorings c of graphs are once again considered but in this case the induced vertex coloringscare neighbor-distinguishing,that is (≠c'(w)fore ery two adiacent vertices u nd v.In this ch apter. d hoth re olors bel ng is sum-defined and the other where)is mul Itiset-defined.Chapter 7is devoted to unrestricted edge colorings of graphs whose colors are elements of Z of integers modulo k that induce a sum-defined,neighbor-distinguishing vertex coloring. In Chap.8.both proper and unrestricted edge colorings are considered,and the vertex colorings are set-defined,using elements of [k]as colors.In Chap.9.the edge colorings are proper and the vertex colorings considered are sum-defined,using elements of[as colors.In these two chapters,the properties of being vertex. distinguishi and neighbor-distinguishing ar both des bed.Cha nds with edge colo which are proper edge colorin use the t ar sum-defined The following table summarizes all types of edge colorings considered in this book and the resulting vertex colorings.In particular,the table describes,in each chapter: 1.the condition placed on the edge coloring 2 the sets from thedkinion which the edge colors are selected. f the colors,and Chapter 1:Introduction Chapter 2:The Irregularity Strength of a Graph Unrestricted Edge Colorings.N.Sum-defined.Vertex-Distinguishing Chapter 3:Modular Sum-defined,Irregular Colorings ined,Vertex-Distinguishing Chapter 4:Se -Defined Irregula Unrestricted Edg ge Colorings.N.Set-definec Vertex-Distinguishing. Chapter 5:Multiset-Defined Irregular Colorings Unrestricted Edge Colorings.N.Multiset-defined.Vertex-Distinguishing Chapter 6:Sum-Defined Neighbor-Distinguishing Colorings Unrestricted Edge Colorings.N.Sum-defined,Neighbor-Distinguishing. Chapter 7:Modular Sum-Defined Neighbor-Distinguishing Colorings Unrestricted Edge Colorings.Sum-defined,Neighbo Distinguishing

vi Preface Chapter 4 also deals with unrestricted edge colorings c of graphs but here the induced vertex coloring is defined so that the color c0 .v/ of a vertex v is the set of colors of its incident edges. In Chap. 5, the emphasis changes from vertex colorings that are set-defined to those that are multiset-defined. In both cases, the induced vertex colorings c0 are vertex-distinguishing. In Chap. 6, unrestricted edge colorings c of graphs are once again considered but in this case the induced vertex colorings c0 are neighbor-distinguishing, that is, c0 .u/ ¤ c0 .v/ for every two adjacent vertices u and v. In this chapter, two vertex colorings c are defined, both where the colors belong to a set Œk, one where c0 .v/ is sum-defined and the other where c0 .v/ is multiset-defined. Chapter 7 is devoted to unrestricted edge colorings of graphs whose colors are elements of Zk of integers modulo k that induce a sum-defined, neighbor-distinguishing vertex coloring. In Chap. 8, both proper and unrestricted edge colorings are considered, and the vertex colorings are set-defined, using elements of Œk as colors. In Chap. 9, the edge colorings are proper and the vertex colorings considered are sum-defined, using elements of Œk as colors. In these two chapters, the properties of being vertex￾distinguishing and neighbor-distinguishing are both described. Chapter 9 ends with a discussion of so-called twin edge colorings, which are proper edge colorings that use the elements of Zk as colors and that induce proper vertex colorings that are sum-defined. The following table summarizes all types of edge colorings considered in this book and the resulting vertex colorings. In particular, the table describes, in each chapter: 1. the condition placed on the edge coloring, 2. the sets from which the edge colors are selected, 3. the definition of the vertex colors, and 4. the property required of the resulting vertex coloring. Chapter 1: Introduction Chapter 2: The Irregularity Strength of a Graph Unrestricted Edge Colorings, N, Sum-defined, Vertex-Distinguishing. Chapter 3: Modular Sum-defined, Irregular Colorings Unrestricted Edge Colorings, Zk, Sum-defined, Vertex-Distinguishing. Chapter 4: Set-Defined Irregular Colorings Unrestricted Edge Colorings, N, Set-defined, Vertex-Distinguishing. Chapter 5: Multiset-Defined Irregular Colorings Unrestricted Edge Colorings, N, Multiset-defined, Vertex-Distinguishing. Chapter 6: Sum-Defined Neighbor-Distinguishing Colorings Unrestricted Edge Colorings, N, Sum-defined, Neighbor-Distinguishing. Chapter 7: Modular Sum-Defined Neighbor-Distinguishing Colorings Unrestricted Edge Colorings, Zk, Sum-defined, Neighbor-Distinguishing.

Preface Chapter 8:Strong Edge Colorings 8.1.Proper Edge Colorings,N.Set-defined,Vertex-Distinguishing. 8.2.Proper and Unrestricted Edge Colorings,N.Set-defined,Vertex-Distinguishing. 8.3.Proper Edge Colorings,N.Set-defined,Neighbor-Distinguishing. Chapter 9:Sum-Defined Colorings by Proper Edge Colorings 9.1.Proper Edge Colorings.N.Sum-defined,Vertex- 9..Proper Edge Colorings. stingu 9.3.Proper Edge Colorings.Z.Sum-defined,Neighbor-Distinguishing Kalamazoo,MI,USA Ping Zhang

Preface vii Chapter 8: Strong Edge Colorings 8.1. Proper Edge Colorings, N, Set-defined, Vertex-Distinguishing. 8.2. Proper and Unrestricted Edge Colorings, N, Set-defined, Vertex-Distinguishing. 8.3. Proper Edge Colorings, N, Set-defined, Neighbor-Distinguishing. Chapter 9: Sum-Defined Colorings by Proper Edge Colorings 9.1. Proper Edge Colorings, N, Sum-defined, Vertex-Distinguishing. 9.2. Proper Edge Colorings, N, Sum-defined, Neighbor-Distinguishing. 9.3. Proper Edge Colorings, Zk, Sum-defined, Neighbor-Distinguishing. Kalamazoo, MI, USA Ping Zhang

Acknowledgements With pleasure.the author thanks Gary Chartrand for the advice and informatior he kindy suppied on may topics described n this book thanks th iewers for able i and s vided with the first draft of this inally,th because of all of you that an improved book resulted. ix

Acknowledgements With pleasure, the author thanks Gary Chartrand for the advice and information he kindly supplied on many topics described in this book. In addition, the author thanks the reviewers for the valuable input and suggestions they provided with the first draft of this manuscript. Finally, the author is very grateful to Razia Amzad, SpringerBriefs editor, for her kindness and encouragement in writing this book. It is because of all of you that an improved book resulted. ix

Contents 1 Introduction...... 1.1 The Origin of Edge Colorings........................ 1.2 Proper Edge Colorings.... 2 1.3 Proper Vertex Colorings 3 2 The Irregularity Strength of a Graph 5 2.1 Defined ity Strength. 2.2 nregularity Strength of Paths and Cycles 1 2.4 Additional Bounds for the Irregularity Strength of a Graph.......... 2 3 Modular Sum-Defined Irregular Colorings................ 3.1 Graceful Graphs..... 5 3.2 Modular Edge-Graceful Graphs 3.3 Non-modular Edge-Graceful Graphs 3.4 Nowhere-Zero Modular Edge-Gra ceful Graphs 40 4 Set-Defined Irregular Colorings 4.1 The Set Irregular Chromatic Index........... 4.2 Complete Graphs and Hypercubes. 4.3 Complete Bipartite Graphs. 49 5 Multiset-Defined Irregular Colorings 5.1 The Multiset Irregular Chromatic Index....... 少 5.2 Regular Graphs 530 omplete Bipartite Graphs t44t4小t4ttt44tte 55 Max-Min Value Problcms 00t。tttt0t4e000。tet。0t。tt500tttt。t0t。ttt。tt0t+。 5638 6 Sum-Defined Neighbor-Distinguishing Colorings........................ 0 6.1 The Sum Distinguishing Index....................... 6.2 The 1-2-3 Conjecture.. 63 6.3 The Multiset Distinguishing Index........ xi

Contents 1 Introduction .................................................................. 1 1.1 The Origin of Edge Colorings .......................................... 1 1.2 Proper Edge Colorings.................................................. 2 1.3 Proper Vertex Colorings ................................................ 3 2 The Irregularity Strength of a Graph ..................................... 5 2.1 Sum-Defined Vertex Colorings: Irregularity Strength ................. 5 2.2 On the Irregularity Strength of Regular Graphs ....................... 8 2.3 The Irregularity Strength of Paths and Cycles ......................... 17 2.4 Additional Bounds for the Irregularity Strength of a Graph .......... 22 3 Modular Sum-Defined Irregular Colorings............................... 31 3.1 Graceful Graphs......................................................... 32 3.2 Modular Edge-Graceful Graphs ........................................ 33 3.3 Non-modular Edge-Graceful Graphs................................... 37 3.4 Nowhere-Zero Modular Edge-Graceful Graphs ....................... 40 4 Set-Defined Irregular Colorings............................................ 43 4.1 The Set Irregular Chromatic Index ..................................... 43 4.2 Complete Graphs and Hypercubes ..................................... 45 4.3 Complete Bipartite Graphs ............................................. 49 5 Multiset-Defined Irregular Colorings...................................... 51 5.1 The Multiset Irregular Chromatic Index ............................... 51 5.2 Regular Graphs.......................................................... 54 5.3 Complete Bipartite Graphs ............................................. 55 5.4 Trees ..................................................................... 56 5.5 Max-Min Value Problems .............................................. 58 6 Sum-Defined Neighbor-Distinguishing Colorings ........................ 61 6.1 The Sum Distinguishing Index ......................................... 61 6.2 The 1-2-3 Conjecture ................................................... 63 6.3 The Multiset Distinguishing Index ..................................... 65 xi

xii Contents 69 21 7.2 Bipartite Graphs........ 7.3 Modular Chromatic Index and Chromatic Number................... 个 8 Strong Edge Colorings of Graphs. 8.1 The Strong Chromatic Index 81 8.2 Binomial Colorings of Graphs 8.3 The Neighbor Strong Chromatic Index. 91 9 Sum-Defined Chromatic Indices 91 The Irregular-Sum Chromatic Index 9.2 The Proper-Sum Chromatic Index .................. 9.3 The Twin Chromatic Index...................... 102 4 Trees.................................................................... 106 References.... 113 117

xii Contents 7 Modular Sum-Defined Neighbor-Distinguishing Colorings ............. 69 7.1 Modular Chromatic Index .............................................. 69 7.2 Bipartite Graphs......................................................... 71 7.3 Modular Chromatic Index and Chromatic Number ................... 77 8 Strong Edge Colorings of Graphs.......................................... 81 8.1 The Strong Chromatic Index ........................................... 81 8.2 Binomial Colorings of Graphs ......................................... 85 8.3 The Neighbor Strong Chromatic Index ................................ 91 9 Sum-Defined Chromatic Indices ........................................... 95 9.1 The Irregular-Sum Chromatic Index ................................... 95 9.2 The Proper-Sum Chromatic Index ..................................... 98 9.3 The Twin Chromatic Index ............................................. 102 9.4 Trees ..................................................................... 106 References......................................................................... 113 Index ............................................................................... 117

List of Figures Fig.2.1 Showing s(K3)=3.. 1 Fig.2.2 An edge coloring of the Petersen graph............................. 9 Fig.2.3 An edge coloring of K4...... 12 Fig.2.4 An edge coloring of Ka Fig.2.5 Constructing the graph H inK 公 g2.6 the proof of for 044.444.4444444.4“444444。小。 Fig.2.7 Edge cole 0ngs0 f Co and C13....... Fig.2.8 Edge c0l0mngs0fC10andC12.......................... 19201 Fig.2.9 Edge colorings of C and Cu.................. Fig.2.10 Illustrating that the inequality in Proposition 2.14 can be strict.... Fig.2.11 Illustrating the equality in Proposition 2.14........ 23 Fig.2.12 An edge coloring of the tree T Fig.2.13 An edge coloring graph 21 Fg2.14 An edge coloring of a connected graph of Fig.3.1 Illustrating B-valuations of C3 and C.. 32 Fig.3.2 Three graphs that are not graceful..... Fig.3.3 Two modular edge-graceful graphs and a non-modular edge-graceful graph........ Fig.3.4 Two modular edge-graceful trees... Fig.3.5 The colorings in Subease 2.I fors3 ands7. 34 Fig.3.6 Two modular edge-graceful colorings of a graph.. 号 The graph set irregular chromatic index Set irregular edge colorings of K and K....... Fig.4.3 A set irregular 4-edge coloring of Ks.. Fig.4.4 A set irregular 3-edge coloring of 2=C4......................... 41

List of Figures Fig. 2.1 Showing s.K3/ D 3 ................................................... 7 Fig. 2.2 An edge coloring of the Petersen graph ............................. 9 Fig. 2.3 An edge coloring of K4;4 ............................................. 12 Fig. 2.4 An edge coloring of K5;5 ............................................. 14 Fig. 2.5 Constructing the graph H in K3.4/ .................................... 15 Fig. 2.6 Edge colorings of Pn in the proof of Theorem 2.12 for 19 Fig. 2.7 Edge colorings of C9 and C13 ........................................ 20 Fig. 2.8 Edge colorings of C10 and C12 ....................................... 21 Fig. 2.9 Edge colorings of C7 and C11 ........................................ 22 Fig. 2.10 Illustrating that the inequality in Proposition 2.14 can be strict .... 23 Fig. 2.11 Illustrating the equality in Proposition 2.14 ......................... 23 Fig. 2.12 An edge coloring of the tree T2 ...................................... 25 Fig. 2.13 An edge coloring of a unicyclic graph G ........................... 27 Fig. 2.14 An edge coloring of a connected graph of size n C 1 .............. 28 Fig. 3.1 Illustrating ˇ-valuations of C3 and C4 ............................... 32 Fig. 3.2 Three graphs that are not graceful ................................... 32 Fig. 3.3 Two modular edge-graceful graphs and a non-modular edge-graceful graph................................................... 33 Fig. 3.4 Two modular edge-graceful trees .................................... 34 Fig. 3.5 The colorings in Subcase 2:1 for s D 3 and s D 7 .................. 39 Fig. 3.6 Two modular edge-graceful colorings of a graph ................... 41 Fig. 4.1 The graph G7 of order 7 with set irregular chromatic index 3 ...... 44 Fig. 4.2 Set irregular edge colorings of K3 and K4 ........................... 46 Fig. 4.3 A set irregular 4-edge coloring of K8 ................................ 46 Fig. 4.4 A set irregular 3-edge coloring of Q2 D C4 ......................... 47 xiii 6 n 10 ............................................................ 19

xiv List of Figures Fig.4.5 a set irregular dge coloring of 4....... 48 Fig.4.6 Showing that si(K2.2)=3 and si(K33)=si(K4.4)=4............ 49 Fig.5.1 A multiset irregular 2-edge coloring.. 52 Fig.5.2 Showing that s(Ps)=3 and mi(Ps)=2.... 2 Fig.5.3 Multiset irregular edge colorings of connected graphs of order 4 Fig.5.4 Fig.6.1 Minimum prope 62 Fig.6.2 a proper sum edge coloring........ Fig.7.1 A modular 3-edge coloring of a graph........... Fig.7.2 A modular 3-edge coloring of the Petersen graph. 10 Fig.8.1 A strong 5-edge coloring of a graph.... Fig.8.2 s of order 7 with st g.&3 The tre romatic index order 15 with ong c omatic index 4... u graphs for k 8 Fig.8.5 Four proper binomial-colored graphs 86 Fig.8.6 A labeled proper 3-binomial-colored graph Fig.8.7 Illustrating a step of the proof of Theorem 8.5................ 导 Fig.8.8 Illustrating the coloring c in the proof of Theorem 8.6 fork4 Fig.8.9 A graph G with (G)=4.. A graph G with xi(G)=5. A graph G with A twin edge 4 900 F1g.9.4 Illustrating Mo.M1.M2.M3 and F1.F2.F3.F for K8............... 5 Fig.9.5 A twin edge 7-coloring of S5.5....................................11

xiv List of Figures Fig. 4.5 Constructing a set irregular 4-edge coloring of Q3 and a set irregular 5-edge coloring of Q4 ................................. 48 Fig. 4.6 Showing that si.K2;2/ D 3 and si.K3;3/ D si.K4;4/ D 4 ............ 49 Fig. 5.1 A multiset irregular 2-edge coloring ................................. 52 Fig. 5.2 Showing that s.P5/ D 3 and mi.P5/ D 2 ............................ 52 Fig. 5.3 Multiset irregular edge colorings of connected graphs of order 4 .............................................................. 53 Fig. 5.4 A step in the proof of Theorem 5.8 .................................. 56 Fig. 6.1 Minimum proper sum colorings of graphs .......................... 62 Fig. 6.2 A multiset neighbor-distinguishing of a tree that is not a proper sum edge coloring........................................... 65 Fig. 7.1 A modular 3-edge coloring of a graph ............................... 70 Fig. 7.2 A modular 3-edge coloring of the Petersen graph................... 70 Fig. 8.1 A strong 5-edge coloring of a graph ................................. 82 Fig. 8.2 The trees of order 7 with strong chromatic index 3 ................. 83 Fig. 8.3 A graph of order 15 with strong chromatic index 4 ................. 83 Fig. 8.4 The k-binomial graphs for k D 2; 3 .................................. 86 Fig. 8.5 Four proper binomial-colored graphs................................ 86 Fig. 8.6 A labeled proper 3-binomial-colored graph ........................ 87 Fig. 8.7 Illustrating a step of the proof of Theorem 8.5 ...................... 88 Fig. 8.8 Illustrating the coloring c in the proof of Theorem 8.6 for k D 4 .............................................................. 90 Fig. 8.9 A graph G with 0 ns.G/ D 4 .......................................... 91 Fig. 9.1 A graph G with 0 is.G/ D 5 .......................................... 96 Fig. 9.2 A graph G with 0 ps.G/ D 4 .......................................... 99 Fig. 9.3 A twin edge 4-coloring of the Petersen graph ....................... 102 Fig. 9.4 Illustrating M0; M1; M2; M3 and F1; F2; F3; F4 for K8 ............... 105 Fig. 9.5 A twin edge 7-coloring of S5;5 ....................................... 111

Chapter 1 Introduction One of the most popular areas of study in graph theory is colorings.This topic can be traced back to the origin of the Four Color Problen and whether it is possible to color the egions of every map with four or fewe two regions having a common b Later it was seen that this problem could be looked at as a problem in graph theory- whethe it is always possible to color the regions of every planar graph (embedded in the plane)so that every two adjacent regions are colored differently.It became known that the Four Color Problem could be solved if it could be solved for all bridgeless cubic planar graphs. 1.1 The Origin of Edge Colorings The Scottish mathematician Tait [76]discovered a unique approach to solve the Four Color Problem.He proved that the edges of a bridgeless cubic planar graph G can be colored with three colors so that every two adjacent edges are colored differently if and only if the regions of G can be colored with four colors so that every two adjacent eareiferey Although Tait's approach ever edtoa solution of the Four Color Problem,he such a 3-coloring of p jo AltEAsps s of G induce an approp e ns of G.The al of induce,in a number of ways,vertex colorings possessing desirable properties. Colors can be objects of any type.While initially,the colors that were used to color the regions of maps were actual colors such as red,blue,green and so on,later it became common to use positive integers for colors as these were simpler and it was easier to keep track of the number of colors being used.Later yet,elements of Zk.the integers modulo k,for somek2.were sometimes used as colors.Subsets or multisets of some set were also used as c lors.Originally,the only requirem Ping Zhang 2015 orings.SpringerBriefs in Mathematics

Chapter 1 Introduction One of the most popular areas of study in graph theory is colorings. This topic can be traced back to the origin of the Four Color Problem and whether it is possible to color the regions of every map with four or fewer colors in such a way that every two regions having a common boundary are assigned different colors. Later it was seen that this problem could be looked at as a problem in graph theory—whether it is always possible to color the regions of every planar graph (embedded in the plane) so that every two adjacent regions are colored differently. It became known that the Four Color Problem could be solved if it could be solved for all bridgeless cubic planar graphs. 1.1 The Origin of Edge Colorings The Scottish mathematician Tait [76] discovered a unique approach to solve the Four Color Problem. He proved that the edges of a bridgeless cubic planar graph G can be colored with three colors so that every two adjacent edges are colored differently if and only if the regions of G can be colored with four colors so that every two adjacent regions are colored differently. Although Tait’s approach never led to a solution of the Four Color Problem, he was able to prove that such a 3-coloring of the edges of G induce an appropriate 4-coloring of the regions of G. The goal of this book is to describe a variety of edge colorings that have been introduced which induce, in a number of ways, vertex colorings possessing desirable properties. Colors can be objects of any type. While initially, the colors that were used to color the regions of maps were actual colors such as red, blue, green and so on, later it became common to use positive integers for colors as these were simpler and it was easier to keep track of the number of colors being used. Later yet, elements of Zk, the integers modulo k, for some k 2, were sometimes used as colors. Subsets or multisets of some set were also used as colors. Originally, the only requirement © Ping Zhang 2015 P. Zhang, Color-Induced Graph Colorings, SpringerBriefs in Mathematics, DOI 10.1007/978-3-319-20394-2_1 1