Graph Coloring Graph Coloring CSE, IIT KGP
K- - coloring coloring K • A A k k- -coloring coloring of G is a labeling of G is a labeling f:V(G) f:V(G) � � {1, {1,… …,k}. ,k}. • – The labels are The labels are colors colors – – The vertices with color – The vertices with color i i are a are a color class color class ↔ y ≠ f(y) x ↔ f(x) ≠ – A A k k - -coloring is coloring is proper proper if if x y implies implies f(x) f(y) – – A graph G is – A graph G is k k- -colorable colorable if it has a proper if it has a proper k k- -coloring coloring χ (G) chromatic number χ – The The chromatic number (G) is the maximum is the maximum k k such such – that G is k k - -colorable colorable that G is χ (G) = k If χ – If (G) = k , then G is , then G is k k- -chromatic chromatic – χ (G) = k χ (H) < k If χ , but χ – If (G) = k , but (H) < k for every proper for every proper subgraph subgraph H H – of G, G, then then G G is is color color- -critical critical or or k k- -critical critical of CSE, IIT KGP
Order of the largest clique Order of the largest clique α (G) denote the Let α • Let (G) denote the independence number independence number of of • ω (G) denote the order of the largest G, and ω (G) denote the order of the largest G, and complete subgraph subgraph of G. of G. complete – χ χ (G) may exceed ω (G). Consider G = C ∨ K (G) may exceed ω 2r+1 ∨ (G). Consider G = C 2r+1 K s – s CSE, IIT KGP
Cartesian Product Cartesian Product • The The Cartesian product Cartesian product of graphs G and H, of graphs G and H, • H, is the graph with vertex set written as G H, is the graph with vertex set written as G V(G) X V(H) specified by putting ( u,v u,v ) ) V(G) X V(H) specified by putting ( ′ ,v ′ ) if and only if u ′ ,v ′ adjacent to ( u ) if and only if adjacent to ( ′ and vv ′∈ E(H), or (1) u = u ′ and vv ′∈ – (1) u = u E(H), or – ′ and ′∈ E(G) (2) v = v ′ uu ′∈ – (2) v = v and uu E(G) – K colorable if and only if G A graph G is m m- - colorable if and only if G K m has A graph G is m has an idependent idependent set of size n(G). set of size n(G). an χ (G H) = max{ χ (G), χ (H) } Also: χ (G H) = max{ χ (G), χ (H) } Also: CSE, IIT KGP
Algorithm Greedy- -Coloring Coloring Algorithm Greedy • The greedy coloring with respect to a vertex The greedy coloring with respect to a vertex • ordering v v 1 ,… …, , v v n of V(G) V(G) is obtained by is obtained by ordering 1 , n of coloring vertices in the order v v 1 ,… …, , v v n , coloring vertices in the order 1 , n , assigning to v v i the smallest indexed color assigning to i the smallest indexed color not already used on its lower- -indexed indexed not already used on its lower neighbors. neighbors. CSE, IIT KGP
Results Results • χ χ (G) ≤ ∆ ∆ (G) + 1 (G) ≤ (G) + 1 • χ (G) = ω (G) If G is an interval graph, then χ (G) = ω • If G is an interval graph, then (G) • ≥ … ≥ d 1 ≥ … ≥ • If a graph G has degree sequence d If a graph G has degree sequence d 1 d n , • n , χ (G) ≤ 1 + max − 1} then χ (G) ≤ , i − 1 + max i min{ d d i 1} then i min{ i , i CSE, IIT KGP
More results More results δ (H) ≥ k − 1 critical graph, then δ (H) ≥ k − • If H is a If H is a k k - -critical graph, then 1 • χ (G) ≤ 1+ δ (H) If G is a graph, then χ (G) ≤ G δ • If G is a graph, then 1+ max max H (H) • ⊆ G H ⊆ • Brooks Theorem: Brooks Theorem: • If G is a connected graph other than a clique If G is a connected graph other than a clique χ (G) ≤ ∆ ∆ (G). or an odd cycle, then χ (G) ≤ (G). or an odd cycle, then CSE, IIT KGP
Mycielski’s Construction Construction Mycielski’s • Mycielski Mycielski found a construction that builds from found a construction that builds from • any given k k - -chromatic triangle chromatic triangle- -free graph G a free graph G a any given ′ . G ′ k+1 - -chromatic triangle chromatic triangle- -free free supergraph supergraph G . k+1 – Given G with vertex set V = {v Given G with vertex set V = {v 1 ,… …, ,v v n }, add vertices – 1 , n }, add vertices U = {u 1 U = {u 1 , ,… …,u ,u n n } and one more vertex w. } and one more vertex w. Beginning with G ′ ′ [V] = G, add edges to make – Beginning with G [V] = G, add edges to make u u i – i adjacent to all of N G (v i ), and then make N(w) = U. adjacent to all of N G (v i ), and then make N(w) = U. ′ . Note that U is an independent set in G ′ . Note that U is an independent set in G CSE, IIT KGP
Critical Graphs Critical Graphs χ (G) > Suppose that G is a graph with χ • (G) > k k and that X,Y and that X,Y • Suppose that G is a graph with is a partition of V(G). If G[X] and G[Y] are k k- - colorable, colorable, is a partition of V(G). If G[X] and G[Y] are then the edge cut [X,Y] has at least k k edges. edges. then the edge cut [X,Y] has at least − 1 edge k − • [Dirac Dirac] ] Every Every k k - -critical graph is critical graph is k 1 edge- -connected. connected. • [ CSE, IIT KGP
Critical Graphs Critical Graphs Suppose S is a set of vertices in a graph G. An S S - - Suppose S is a set of vertices in a graph G. An component of G is an induced subgraph subgraph of G whose of G whose component of G is an induced vertex set consists of S S and the vertices of a and the vertices of a vertex set consists of − S. component of G − S. component of G • If G is k k - -critical, then G has no critical, then G has no cutset cutset of vertices of vertices • If G is inducing a clique. In particular, if G has a cutset cutset inducing a clique. In particular, if G has a S={x,y}, then x and y are not adjacent and G has an S={x,y}, then x and y are not adjacent and G has an χ (H + ≥ k component H such that χ ) ≥ S - -component H such that (H + xy xy) k . . S CSE, IIT KGP
Chromatic Recurrence Chromatic Recurrence χ (G; The function χ (G; k k ) counts the mappings ) counts the mappings f: f: V(G) V(G) � � [k] [k] The function that properly color G from the set [k] = {1,… …,k}. In this ,k}. In this that properly color G from the set [k] = {1, definition, the k k - -colors need not all be used, and colors need not all be used, and definition, the permuting the colors used produces a different permuting the colors used produces a different coloring. coloring. ∈ E(G), then e ∈ • If G is a simple graph and If G is a simple graph and e E(G), then • χ (G; k) = χ (G − e − χ χ (G . e χ (G; k) = χ (G − ; k) − (G . e ; k) e ; k) ; k) CSE, IIT KGP
Line Graphs Line Graphs The line graph line graph of G, written of G, written L(G) L(G) , is a simple graph , is a simple graph The ∈ E(L(G)) ef ∈ whose vertices are the edges of G, with ef E(L(G)) whose vertices are the edges of G, with when e e and and f f share a vertex of G. share a vertex of G. when • An Eulerian Eulerian circuit in G yields a spanning cycle in circuit in G yields a spanning cycle in • An L(G). The converse need not hold L(G). The converse need not hold • A matching in G is an independent set in L(G); we • A matching in G is an independent set in L(G); we α′ (G) = α (L(G)) have α′ (G) = α (L(G)) have CSE, IIT KGP
Edge Coloring Edge Coloring A k k- -edge edge- -coloring coloring of G is a labeling of G is a labeling f f : E(G) : E(G) � � [k] [k] A – The labels are The labels are colors colors – – The set of edges with one color is a The set of edges with one color is a color class. color class. – – A A k k- - edge edge- -coloring is coloring is proper proper if edges sharing a if edges sharing a – vertex have different colors; equivalently, each vertex have different colors; equivalently, each color class is a matching color class is a matching – A graph is A graph is k k- -edge edge- -colorable colorable if it has a proper if it has a proper k k - - – edge- -coloring coloring edge χ′ (G) of a loop number χ′ – The The edge edge- -chromatic chromatic- -number (G) of a loop- -less less – graph G is the least k k such that G is such that G is k k- -edge edge- - graph G is the least colorable colorable CSE, IIT KGP
Results Results χ′ (G) ≥ ∆ ∆ (G). χ′ (G) ≥ • (G). • χ′ (G) ≤ 2 ∆ (G) − 1. less graph, then χ′ (G) ≤ 2 ∆ (G) − • If G is a loop- -less graph, then 1. • If G is a loop χ′ (G) = ∆ (G). If G is bipartite, then χ′ (G) = ∆ • (G). • If G is bipartite, then ∆ (G) A regular graph G has a ∆ (G)- -edge coloring if and edge coloring if and A regular graph G has a only if it decomposes into 1- -factors. We say that G is factors. We say that G is only if it decomposes into 1 1- -factorable. factorable. 1 ∆ has a Every simple graph with maximum degree ∆ • has a • Every simple graph with maximum degree ∆ +1 proper ∆ +1- -edge edge- -coloring. coloring. proper CSE, IIT KGP
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