TY - JOUR

T1 - List total colorings of series-parallel graphs

AU - Zhou, Xiao

AU - Matsuo, Yuki

AU - Nishizeki, Takao

PY - 2003

Y1 - 2003

N2 - A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, in such a way that no two adjacent or incident elements receive the same color. Let L(x) be a set of colors assigned to each element x of G. Then a list total coloring of G is a total coloring such that each element x receives a color contained in L(x). The list total coloring problem asks whether G has a list total coloring. In this paper, we first show that the list total coloring problem is NP-complete even for series-parallel graphs. We then give a sufficient condition for a series-parallel graph to have a list total coloring, that is, we prove a theorem that any series-parallel graph G has a list total coloring if |L/(v)| ≥ min{5, Δ + 1} for each vertex v and |L(e)| ≥ max{5, d(v) + 1, d(w) + 1} for each edge e = vw, where Δ is the maximum degree of G and d(v) and d(w) are the degrees of the ends v and w of e, respectively. The theorem implies that any series-parallel graph G has a total coloring with Δ + 1 colors if Δ ≥ 4. We finally present a linear-time algorithm to find a list total coloring of a given series-parallel graph G if G satisfies the sufficient condition.

AB - A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, in such a way that no two adjacent or incident elements receive the same color. Let L(x) be a set of colors assigned to each element x of G. Then a list total coloring of G is a total coloring such that each element x receives a color contained in L(x). The list total coloring problem asks whether G has a list total coloring. In this paper, we first show that the list total coloring problem is NP-complete even for series-parallel graphs. We then give a sufficient condition for a series-parallel graph to have a list total coloring, that is, we prove a theorem that any series-parallel graph G has a list total coloring if |L/(v)| ≥ min{5, Δ + 1} for each vertex v and |L(e)| ≥ max{5, d(v) + 1, d(w) + 1} for each edge e = vw, where Δ is the maximum degree of G and d(v) and d(w) are the degrees of the ends v and w of e, respectively. The theorem implies that any series-parallel graph G has a total coloring with Δ + 1 colors if Δ ≥ 4. We finally present a linear-time algorithm to find a list total coloring of a given series-parallel graph G if G satisfies the sufficient condition.

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U2 - 10.1007/3-540-45071-8_19

DO - 10.1007/3-540-45071-8_19

M3 - Article

AN - SCOPUS:33750694342

SN - 0302-9743

VL - 2697

SP - 172

EP - 181

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

ER -