Abstract
In online list coloring [introduced by Zhu (Electron J Comb 16(1):#R127, 2009) and Schauz (Electron J Comb 16:#R77, 2009)], on each round the set of vertices having a particular color in their lists is revealed, and the coloring algorithm chooses an independent subset of this set to receive that color. For a graph \(G\) and a function \(f:\,V(G)\rightarrow {\mathbb N}\), the graph is \(f\)-paintable if there is an algorithm to produce a proper coloring when each vertex \(v\) is allowed to be presented at most \(f(v)\) times. The sum-paintability of \(G\), denoted \(\chi _{sp}(G)\), is \(\min \{\sum f(v):\,G\) is \(f\)-paintable\(\}\). Basic results include \(\chi _{sp}(G)\le |V(G)|+|E(G)|\) for every graph \(G\) and \(\chi _{sp}(G)=(\sum _{i=1}^{k} \chi _{sp}(H_i))-(k-1)\) when \(H_{1},\ldots ,H_{k}\) are the blocks of \(G\). Also, adding an ear of length \(\ell \) to \(G\) adds \(2\ell -1\) to the sum-paintability, when \(\ell \ge 3\). Strengthening a result of Berliner et al., we prove \(\chi _{sp}(K_{2,r})=2r + \min \{l+m:\,lm>r\}\) . The generalized theta-graph \(\Theta _{\ell _{1},\ldots ,\ell _{k}}\) consists of two vertices joined by internally disjoint paths of lengths \(\ell _{1},\ldots ,\ell _{k}\). A book is a graph of the form \(\Theta _{1,2,\dots ,2}\), denoted \(B_r\) when there are \(r\) internally disjoint paths of length 2. We prove \(\chi _{sp}(B_r)=2r + \min _{l,m\in \mathbb {N}}\{l+m:\,m(l-m)+{m\atopwithdelims ()2}>r\}\). We use these results to determine the sum-paintability for all generalized theta-graphs.
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Berliner, A., Bostelmann, U., Brualdi, R.A., Deaett, L.: Sum list coloring graphs. Graphs Comb. 22, 173–183 (2006)
Carraher, J., Loeb, S., Mahoney, T., Puleo, G.J., Tsai, M., West, D.B.: Three topics in online list coloring J. Comb. 5, 115–130 (2014)
Erdős, P., Rubin, A.L., Taylor, H.: Choosability in graphs. Proc. West Coast Conf. Comb. Graph Theory Comput. (Humboldt State Univ. Arcata Calif. Congr. Numer. 1979) 26, 125–157 (1980)
Heinold, B.: Sum list coloring and choosability. PhD thesis, Lehigh University (2006)
Isaak, G.: Sum list coloring \(2\times n\) arrays. Electron. J. Comb. 9, #N8 (2002)
Isaak, G.: Sum list coloring block graphs. Graphs Comb. 20, 499–506 (2004)
Kim, S.-J., Kwon, Y., Liu, D.D., Zhu, X.: On-line list colouring of complete multipartite graphs. Electron. J. Comb. 19, Paper #P41, 13 (2012)
Kozik, J., Micek, P., Zhu, X.: Towards an on-line Ohba’s conjecture Eur. J. Comb. 36, 110–121 (2014)
Lastrina, M.: List-coloring and sum-list-coloring problems on graphs. Graduate theses and Dissertations, Paper 12376 (2012)
Mahoney, T., Tomlinson, C., Wise, J.I.: Families of online sum-choice-greedy graphs. Graphs Comb. (to appear)
Schauz, U.: Mr. Paint and Mrs. Correct. Electron. J. Comb. 16, #R77 (2009)
Vizing, V.G.: Vertex colorings with given colors (Russian). Diskret. Anal. 29, 3–10 (1976)
Zhu, X.: On-line list colouring of graphs. Electron. J. Comb. 16(1), #R127 (2009)
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Research of D. B. West supported by NSA Grant H98230-10-1-0363 and by Recruitment Program of Foreign Experts, 1000 Talent Plan, State Administration of Foreign Experts Affairs, China. Research of J.M. Carraher supported by NSF Grant DMS 09-14815. Research of T. Mahoney and G. J. Puleo supported by by NSF Grant DMS 08-38434, “EMSW21-MCTP: Research Experience for Graduate Students”. The primary affiliation of D.B. West is Zhejiang Normal University.
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Carraher, J.M., Mahoney, T., Puleo, G.J. et al. Sum-Paintability of Generalized Theta-Graphs. Graphs and Combinatorics 31, 1325–1334 (2015). https://doi.org/10.1007/s00373-014-1441-1
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DOI: https://doi.org/10.1007/s00373-014-1441-1