Flots et couvertures par des cycles dans les graphes et les matroiedes

CNRS T 57179 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueSIGLEFRFranc

On star edge colorings of bipartite and subcubic graphs

A star edge coloring of a graph is a proper edge coloring with no 2-colored path or cycle of length four. The star chromatic index chi(st)(G) of G is the minimum number t for which G has a star edge coloring with t colors. We prove upper bounds for the star chromatic index of bipartite graphs G where all vertices in one part have maximum degree 2 and all vertices in the other part has maximum degree b. Let k be an integer (k &amp;gt;= 1); we prove that if b = 2k + 1, then chi(st)(G) &amp;lt;= 3k + 2; and if b = 2k, then chi(st)(G) &amp;lt; 3k; both upper bounds are sharp. We also consider complete bipartite graphs; in particular we determine the star chromatic index of such graphs when one part has size at most 3, and prove upper bounds for the general case. Finally, we consider the well-known conjecture that subcubic graphs have star chromatic index at most 6; in particular we settle this conjecture for cubic Halin graphs. (C) 2021 The Authors. Published by Elsevier B.V.Funding Agencies|Swedish Research CouncilSwedish Research CouncilEuropean Commission [2017-05077]; French ANRFrench National Research Agency (ANR) [ANR-17-CE40-0022]</p

Recognizing Recursive Circulant Graphs Abstract (Extended Abstract)

Recursive circulant graphs G(N�d) have been introduced in 1994 by Park and Chwa [PC94] as a new topology for interconnection networks. Recursive circulant graphs G(N�d) are circulant graphs with N nodes and with jumps of powers of d. A subfamily of recursive circulant graphs (more precisely, G(2 k � 4)) is of same order and degree than the hypercube of dimension k, with sometimes better parameters, such as diameter [PC94,GMR98]. Embeddings among recursive circulant graphs, hypercubes and Knodel graphs of order 2 k have also been studied in [PC,FR98b]. Here, following a question raised in [CFG99], we give, thanks to a sharp structural analysis of such graphs, an O(cd m+2 (2m) d) algorithm to determine if a given graph is a recursive circulant graph of the form G(cd m �d), for any d 3, except in the case where c is even while d is odd. Notably, in the case where d = O(1), this gives an O(N (log(N)) O(1) ) algorithm, with N = cd m.Moreover, applying this algorithm to recursive circulant graphs G(2 k � 4) gives us an O(2 k k 4) recognition algorithm for such graphs