Proper connection number of graphs
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1 Proper connection number of graphs Ruxandra MARINESCU-GHEMECI University of Bucharest Mihai Patrascu Seminar in Theoretical Computer Science Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 1 / 34
2 Proper colorings Four Color Theorem Figure: Map coloring Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 2 / 34
3 Proper colorings Four Color Theorem Figure: Map coloring - Planar graph proper coloring Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 2 / 34
4 Proper colorings Proper colorings four color map problem [Appel and Haken (1989)] many applications - avoid conflicts register allocation scheduling problems interference, security in communication networks Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 3 / 34
5 Proper colorings Proper colorings Definition Proper (vertex) coloring c : V (G) {1, 2,..., k} c(x) c(y), xy E(G) Chromatic number of G χ(g)= minimum k such that G has a proper coloring with k colors: c : V (G) {1, 2,..., k} Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 4 / 34
6 Proper colorings Proper edge colorings Definition Proper edge coloring c : E(G) {1, 2,..., k} c(e) c(f ), e, f E(G) adjacent (that have a common vertex) Edge chromatic number of G χ (G)= minimum k such that G has a proper edge coloring with k colors: c : E(G) {1, 2,..., k} Figure: Proper edge-coloring - interference graph Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 5 / 34
7 Proper colorings Proper path colorings Relaxed constrains Not necessary to impose constrains on the colors for all pairs of adjacent edges/vertices in order avoid conflicts, but: assure paths between any pair of vertices on which communication is safe Possible advantages: less colors, algorithms Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 6 / 34
8 Proper colorings Proper path colorings Relaxed constrains Not necessary to impose constrains on the colors for all pairs of adjacent edges/vertices in order avoid conflicts, but: assure paths between any pair of vertices on which communication is safe Possible advantages: less colors, algorithms Proper path coloring - proper connection number Borozan et al. (2012) Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 6 / 34
9 Proper colorings Proper path colorings path = lanţ elementar trail = lanţ simplu walk = lanţ Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 7 / 34
10 Proper colorings Proper path colorings Definition (Borozan et al. (2012); Andrews et al. (2016)) c : E(G) {1, 2,..., k} proper path P: c E(P) proper edge-coloring proper connection number of G: pc(g): minimum k such that there is a coloring c : E(G) {1, 2,..., k} that makes G proper connected: for every pair u,v of vertices exists a proper path from u to v. Related to rainbow coloring - [Chartrand et al. (2008)] Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 8 / 34
11 Proper connection number Known results Survey - [Li and Magnant (2015)] Combinatorial results: Existence problems [Andrews et al. (2016)] 3 2 difference between chromatic number and pc can be arbitrary large 4 1 Connection to structure properties of graph Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 9 / 34
12 Proper connection number Known results Survey - [Li and Magnant (2015)] Combinatorial results: Existence problems [Andrews et al. (2016)] difference between chromatic number and pc can be arbitrary large Connection to structure properties of graph Extremal graphs [Laforge et al. (2016)] graphs with pc(g) = m 1, m 2 graphs with pc = 2-3-connected, 2-connected with diameter 2, bipartite bridgeless Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 9 / 34
13 Proper connection number Classes of graphs - Bipartite graphs sufficient conditions to have proper connection number 2 [Borozan et al. (2012); Huang et al. (2015, 2016)] Complete characterization [Ducoffe et al. (2017)] Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 10 / 34
14 Proper connection number Proper connection number Remark pc(g) = 1 G is complete. Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 11 / 34
15 Bipartite graphs Definition Bipartite graph has a proper coloring with 2 colors. Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 12 / 34
16 Bipartite graphs Remark 1 G is bipartite G contains no odd cycle. 2 Let G be a bipartite graph and u, v V (G). Then the lengths of all walks between u and v have the same parity. 3 If G is a bipartite graph with at least 3 vertices, G is not complete. Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 13 / 34
17 Bipartite graphs Remark 1 G is bipartite G contains no odd cycle. 2 Let G be a bipartite graph and u, v V (G). Then the lengths of all walks between u and v have the same parity. 3 If G is a bipartite graph with at least 3 vertices, G is not complete. Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 13 / 34
18 Bipartite graphs Remark 1 G is bipartite G contains no odd cycle. 2 Let G be a bipartite graph and u, v V (G). Then the lengths of all walks between u and v have the same parity. 3 If G is a bipartite graph with at least 3 vertices, G is not complete. Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 13 / 34
19 Bipartite graphs with proper connection number 2 Cases. Necessary notions 1 G is 2-(edge) connected [Borozan et al. (2012)] cut-edge, cut-vertex 2-(edge) connected graph ear decomposition 2 G arbitrary (connected) - necessary and sufficient conditions block, bridge-block bridge-block tree Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 14 / 34
20 Connectivity Definition Let G be a connected graph. v V (G) cut vertex (punct critic, de articulaţie) G v is not connected e E(G) bridge (punte, muchie critică) G e is not connected 2-connected graph = has no cut-vertices 2-edge connected (bridgeless) graph = has no bridges Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 15 / 34
21 Connectivity Definition Let G be a connected graph. v V (G) cut vertex (punct critic, de articulaţie) G v is not connected e E(G) bridge (punte, muchie critică) G e is not connected 2-connected graph = has no cut-vertices 2-edge connected (bridgeless) graph = has no bridges Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 15 / 34
22 Ear decomposition What is an ear + ear decomposition? Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 16 / 34
23 Ear decomposition What is an ear + ear decomposition? Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 16 / 34
24 Ear decomposition What is an ear + ear decomposition? Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 16 / 34
25 Ear decomposition What is an ear + ear decomposition? Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 16 / 34
26 Ear decomposition What is an ear + ear decomposition? Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 16 / 34
27 Ear decomposition What is an ear + ear decomposition? Figure: Open ear decomposition Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 16 / 34
28 Ear decomposition Figure: Closed ear decomposition Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 17 / 34
29 Ear decomposition Definition Ear of G = maximal path whose internal vertices have degree 2 in G -open ear = distinct extremities -closed ear = can have equal extremities (=can be a cycle) Ear decomposition (open/closed) of G = decomposition P 0,..., P k such P 0 is a cycle P i is an ear (open/closed) of P 0... P i Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 18 / 34
30 2- connected graphs - Ear decomposition Theorem (Whitney) A graph is 2-connected has an open ear decomposition. Furthermore, every cycle in G is the initial cycle in some ear decomposition. Algorithms for finding ear decomposition - O( V + E ) Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 19 / 34
31 2- connected graphs - Ear decomposition Theorem (Whitney) A graph is 2-connected has an open ear decomposition. Furthermore, every cycle in G is the initial cycle in some ear decomposition. Algorithms for finding ear decomposition - O( V + E ) Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 19 / 34
32 2- connected graphs - Ear decomposition Theorem (Whitney) A graph is 2-connected has an open ear decomposition. Furthermore, every cycle in G is the initial cycle in some ear decomposition. Algorithms for finding ear decomposition - O( V + E ) Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 19 / 34
33 2- connected graphs - Ear decomposition Theorem (Whitney) A graph is 2-connected has an open ear decomposition. Furthermore, every cycle in G is the initial cycle in some ear decomposition. Algorithms for finding ear decomposition - O( V + E ) Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 19 / 34
34 2-edge connected graphs - Ear decomposition Theorem (Whitney) A graph is 2-edge connected has a closed ear decomposition. Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 20 / 34
35 Bipartite graphs with proper connection number 2 Notion and notations P = [u 1,..., u s ] length l(p) = s 1 odd/even path = of odd/even length P u i uj P 1 = [u s,..., u 1] Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 21 / 34
36 Bipartite graphs with proper connection number 2 Notion and notations P = [u 1,..., u s ] length l(p) = s 1 odd/even path = of odd/even length u i P uj P 1 = [u s,..., u 1] Let c and edge coloring start(p) = c(u 1u 2) end(p) = c(u s 1u s) Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 21 / 34
37 Bipartite graphs with proper connection number 2 2-edge connected bipartite graphs Theorem (Borozan et al. (2012)) Let G be a graph. If G is bipartite and 2-connected then pc(g) = 2 and there exists a 2-coloring of G that makes it properly connected with the following strong property. For any pair of vertices u, v there exists two paths P 1, P 2 between them such start(p 1 ) start(p 2 ) and end(p 1 ) end(p 2 ). u v Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 22 / 34
38 Bipartite graphs with proper connection number 2 Definition Block (2-connected subgraphs, biconnected component) of G - maximal 2-connected subgraph (without cut vertices) block-cut tree Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 23 / 34
39 Bipartite graphs with proper connection number 2 Definition Block (2-connected subgraphs, biconnected component) of G - maximal 2-connected subgraph (without cut vertices) block-cut tree Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 23 / 34
40 Bipartite graphs with proper connection number 2 Definition Block (2-connected subgraphs, biconnected component) of G - maximal 2-connected subgraph (without cut vertices) block-cut tree Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 23 / 34
41 Bipartite graphs with proper connection number 2 Definition Block (2-connected subgraphs, biconnected component) of G - maximal 2-connected subgraph (without cut vertices) block-cut tree Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 23 / 34
42 Bipartite graphs with proper connection number 2 Definition Bridge-block of G = connected component of G minus all bridges Bridge-block tree - vertices corresponding to bridge-blocks - edges corresponding to bridges Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 24 / 34
43 Bipartite graphs with proper connection number 2 Definition Bridge-block of G = connected component of G minus all bridges Bridge-block tree - vertices corresponding to bridge-blocks - edges corresponding to bridges Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 24 / 34
44 Bipartite graphs with proper connection number 2 Definition Bridge-block of G = connected component of G minus all bridges Bridge-block tree - vertices corresponding to bridge-blocks - edges corresponding to bridges Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 24 / 34
45 Bipartite graphs with proper connection number 2 Definition Bridge-block of G = connected component of G minus all bridges Bridge-block tree - vertices corresponding to bridge-blocks - edges corresponding to bridges Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 24 / 34
46 Bipartite graphs with proper connection number 2 What if the graph has bridges? Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 25 / 34
47 Bipartite graphs with proper connection number 2 What if the graph has bridges? Remark pc(g) b(g) = maximum number of bridges incident in a vertex pc(g) 2 = b(g) 2 Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 25 / 34
48 Bipartite graphs with proper connection number 2 What if the graph has bridges? Remark pc(g) b(g) = maximum number of bridges incident in a vertex pc(g) 2 = b(g) 2? Any bipartite graph with b(g) 2 has pc(g) 2? Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 25 / 34
49 Bipartite graphs? Any bipartite graph with b(g) 2 has pc(g) 2? Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 26 / 34
50 Bipartite graphs? Any bipartite graph with b(g) 2 has pc(g) 2? No Lemma Let G = (V, E) be a connected graph, B be a bridge-block of G that is bipartite. If B is incident to at least three bridges then pc(g) 3. v 0 u 0 v 1 u 1 u 2 v 2 u 3 Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 26 / 34
51 Bipartite graphs Assume by contradiction that G has proper path coloring with 2 colors. v 0 b 0 u 0 v 1 u 1 u 2 v 2 b 1 b 2 u 3 Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 27 / 34
52 Bipartite graphs paths of same parity v 0 b 0 u 0 v 1 u 1 u 2 v 2 b 1 b 2 u 3 Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 27 / 34
53 Bipartite graphs paths of same parity v 0 b 0 u 0 v 1 u 1 u 2 v 2 b 1 b 2 u 3 Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 27 / 34
54 Bipartite graphs paths of same parity v 0 b 0 u 0 v 1 u 1 u 2 v 2 b 1 b 2 u 3 even length Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 27 / 34
55 Bipartite graphs Theorem Let G = (V, E) be a connected bipartite graph. We have pc(g) 2 if and only if the bridge-block tree of G is a path. Furthermore, if pc(g) 2, then such a coloring can be computed in linear-time. B 0 B 1 B 2 B 3 B 4 Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 28 / 34
56 Bipartite graphs If bridge-block tree of G is a path = linear ordering B 0, B 1,..., B l over the bridge-blocks. Color blocks in this order (with strong property) B 0 B 1 B 2 B 3 B 4 Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 29 / 34
57 Bipartite graphs If bridge-block tree of G is a path = linear ordering B 0, B 1,..., B l over the bridge-blocks. Color B 0 (with strong property) B 0 B 1 B 2 B 3 B 4 Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 29 / 34
58 Bipartite graphs If bridge-block tree of G is a path = linear ordering B 0, B 1,..., B l over the bridge-blocks. Color bridge between B 0 and B 1 arbitrary B 0 B 1 B 2 B 3 B 4 Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 29 / 34
59 Bipartite graphs If bridge-block tree of G is a path = linear ordering B 0, B 1,..., B l over the bridge-blocks. Color B 1 (with strong property) B 0 B 1 B 2 B 3 B 4 Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 29 / 34
60 Bipartite graphs If bridge-block tree of G is a path = linear ordering B 0, B 1,..., B l over the bridge-blocks. Color bridge between B i and B i+1 according to color of bridge between B i 1 and B i and parity of paths length B 0 B 1 B 2 B 3 B 4 Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 29 / 34
61 Bipartite graphs If bridge-block tree of G is a path = linear ordering B 0, B 1,..., B l over the bridge-blocks. Color bridge between B i and B i+1 according to color of bridge between B i 1 and B i and parity of paths length B 0 B 1 B 2 B 3 B 4 Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 29 / 34
62 Bipartite graphs If bridge-block tree of G is a path = linear ordering B 0, B 1,..., B l over the bridge-blocks. Color bridge between B i and B i+1 according to color of bridge between B i 1 and B i and parity of paths length B 0 B 1 B 2 B 3 B 4 Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 29 / 34
63 Bipartite graphs If bridge-block tree of G is a path = linear ordering B 0, B 1,..., B l over the bridge-blocks. Color bridge between B i and B i+1 according to color of bridge between B i 1 and B i and parity of paths length B 0 B 1 B 2 B 3 B 4 Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 29 / 34
64 Bipartite graphs If bridge-block tree of G is a path = linear ordering B 0, B 1,..., B l over the bridge-blocks. Color bridge between B i and B i+1 according to color of bridge between B i 1 and B i and parity of paths length B 0 B 1 B 2 B 3 B 4 Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 29 / 34
65 3-connnected graphs Proposition Let G be a graph. If G is 3-connected and noncomplete, then pc(g) = 2 and there exists a 2-edge-coloring of G that makes it proper connected and has the strong property. Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 30 / 34
66 3-connnected graphs Proposition Let G be a graph. If G is 3-connected and noncomplete, then pc(g) = 2 and there exists a 2-edge-coloring of G that makes it proper connected and has the strong property. Proof. Any 3-connected graph has a spanning 2-connected bipartite subgraph [Paulraja and Magnant (1993)] Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 30 / 34
67 Open problems NP-completness results and algorithms for undirected graphs Directed graphs Other type of chromatic connectivity Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 31 / 34
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69 Bibliography E. Andrews, C. Lumduanhom, E. Laforge, and P. Zhang. On proper-path colorings in graphs. J. Comb. Math. and Comb. Comp., 97: , K. Appel and W. Haken. Every Planar Map is Four-Colorable. Contemporary Mathematics, V. Borozan, S. Fujita, A. Gerek, C. Magnant, Y. Manoussakis, L. Montero, and Z. Tuza. Proper connection of graphs. Discrete Mathematics, 312(17): , G. Chartrand, G. L. Johns, K. A. McKeon, and P. Zhang. Rainbow connection in graphs. Mathematica Bohemica, 133(1):85 98, G. Ducoffe, R. Marinescu-Ghemeci, and A. Popa. On the (di) graphs with (directed) proper connection number two F. Huang, X. Li, and S. Wang. Proper connection number and 2-proper connection number of a graph. arxiv preprint arxiv: , F. Huang, X. Li, Z. Qin, and C. Magnant. Minimum degree condition for proper connection number 2. Theoretical Computer Science, Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 33 / 34
70 Bibliography E. Laforge, C. Lumduanhom, and P. Zhang. Characterizations of graphs having large proper connection numbers. Discuss. Math. Graph Theory, 36(2): , X. Li and C. Magnant. Properly colored notions of connectivity-a dynamic survey. Theory and Applications of Graphs, (1):2, P. Paulraja and C. Magnant. A characterization of hamiltonian prisms. J. Graph Theory, 2(17):161171, Ruxandra MARINESCU-GHEMECI (UB) Proper connection number of graphs 34 / 34
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